1, for *x* = (*z*, ⋯, *z*) ∈ Δ*S* with *z* ∈ *M*, the normal bundle *ν*Δ*S*, *x* of Δ*S* in *M**S* at *x* can be identified with the quotient space (*T**z**M*)*S*/*T**z**M*. For *u**S* = (*u**i*)*i* ∈ *S* ∈ (*T**z**M*)*S*, if all components *u**i* are identical, then [*u**S*] = 0 ∈ (*T**z**M*)*S*/*T**z**M*. For each element [*u**S*] ∈ (*T**z**M*)*S*/*T**z**M*, there exists a unique representative *u**S* = (*u**i*)*i* ∈ *S* ∈ (*T**z**M*)*S* such that ∑*i* ∈ *S**u**i* = 0 in *T**z**M*. We can also regard such vector $u\\_S \\in (T\\_z M)^{\\underline{n}}$ by setting *u**j* = 0 for *j* ∉ *S*.\nNote that R+\\* = ]0, + ∞] acts on (*T**z**M*)*S*/*T**z**M* by the diagonal rescalling on all the components. So that we have the identification $S\\nu\\_{\\Delta\\_S}\\simeq {\\mathbb{R}}^\\*\\_+\\backslash \\big((T\\_z M)^S/T\\_zM - \\{0\\}\\big)$. Then naturally, any nonzero [*u**S*] ∈ (*T**z**M*)*S*/*T**z**M* corresponds to an element in *S**ν*Δ*S*, which is still denoted by [*u**S*] if there is no confusion.\nLet *q* : *B*ℓ(*M**S*, Δ*S*) → *M**S* denote the obvious projection. We always use *x**B*, *S* to denote a point in *B*ℓ(*M**S*, Δ*S*), such that if *x**S* = *q*(*x**B*, *S*) ∉ Δ*S*, then *x**B*, *S* = *x**S*; otherwise, *x**B*, *S* = (*x**S*, [*u**S*]) where *x**S* = (*z*, ⋯, *z*) ∈ Δ*S*, 0 ≠ [*u**S*] ∈ (*T**z**M*)*S*/*T**z**M*.\nAs a point set, we have a characterization of *C**n*(*M*) as a subset of B: *C**n*(*M*) are consisting of all points (*x*, {*x**B*, *S*, ∣*S*∣ ≥ 2}) in B which satisfy the following two conditions:\n* *x**S* = *q*(*x**B*, *S*) = *x*∣*S*, for $S\\subset \\underline{n}$, ∣*S*∣ > 1, where *x*∣*S* = (*x**i*)*i* ∈ *S* ∈ *M**S* with *x* = (*x*1, ⋯, *x**n*) ∈ *M**n*.\n* For any subset *S* (∣*S*∣ ≥ 3) with *x**S* ∈ Δ*S*, write *x**B*, *S* = (*x**S*, [*u**S*]), then for each subset *S*ʹ ⊂ *S* with ∣*S*ʹ∣ ≥ 2, if *S*ʹ-components of *u**S* are not all equal, we have *x**B*, *S*ʹ = (*x**S*ʹ, [*u**S*∣*S*ʹ]).\nSet $\\mathbf{S}\\_n=\\{S\\subset \\underline{n}\\;|\\; |S|\\geq 2\\}$.\nA subset S ⊂ **S***n* is called nested if any two elements *S*1, *S*2 ∈ S are either disjoint or else one contains the other.\nFor a nested subset S ⊂ **S***n*, the open strata *M*(S)∘ of *C**n*(*M*) is defined as follows, it consists of the points (*x*, {*x**B*, *S*, ∣*S*∣ ≥ 2}) ∈ *C**n*(*M*) such that\n* *x*∣*S* ∈ Δ*S* if and only if *S* ⊂ *S*ʹ for some *S*ʹ ∈ S.\n* For *S*ʹ (∣*S*ʹ∣ > 1) with a minimal element *S* ∈ S such that *S*ʹ ⊂ *S*, then [*u**S*ʹ] = [*u**S*∣*S*ʹ] in *x**B*, *S*ʹ.\n* For *S*1, *S*2 ∈ S such that $S\\_1\\subsetneq S\\_2$, then all *S*1-components of *u**S*2 are all equal.\nIn, Axelrod and Singer showed the following facts:\n* *M*(S)∘ is a smooth (noncompact) manifold of dimension 3*n* − ∣S∣, in particular, *M*(∅)∘ = Conf*n*(*M*).\n* The closed strata *M*(S), defined as the closure of *M*(S)∘ in *C**n*(*M*), is given as follows \n*M*(S) = ∪ T ⊃ S*M*(T)∘ \n where T runs over all nested subsets of **S***n* containing S.\n* We have\n \n*C**n*(*M*) = ∪ S nested*M*(S)∘.\n* For the strata of *C**n*(*M*) as manifold with corners, we have for *k* = 0, 1, ⋯, 3*n*,\n \n*S**k**C**n*(*M*) = ∪ S, ∣*S*∣ = *k**M*(S)∘. \nThis way, we see that the interior of ∂\\**C**n*(*M*) is given by all the single sets *S* of $\\underline{n}$ with ∣*S*∣ ≥ 2.\nThe following proposition allows us to define properly the fibre-wise integration for the projections between compactified configuration spaces.\n[prop:subme] For *n* ≥ 2, for 1 ≤ *i* ≠ *j* ≤ *n*, the projection\n \n*p**i**j* : Conf*n*(*M*) ∋ (*x*1, …, *x**n*) ↦ (*x**i*, *x**j*) ∈ Conf2(*M*) \ninduces a surjective submersion of manifolds with corners\n \n*C**n*(*M*) → *C*2(*M*) \nwhich is still denoted by *p**i**j*.\nWithout loss of generality, we may assume (*i*, *j*) = (1, 2). It is clear that the points in *M*(S)∘ ⊂ *C**n*(*M*) with S containing no {1, 2} map to the points in Conf2(*M*). For such a point, we can verify directly the condition that *p**i**j* is submersion. The points in *M*(S)∘ ⊂ *C**n*(*M*) with S containing {1, 2} map to the points in ∂*C*2(*M*), then the computation for the tangent maps is to consider the projection between the unit spheres of different dimensions, and shows the same conclusion for it being the submersion. This way, we finish the proof.\nFraming of closed orientable 3-manifolds\n----------------------------------------\nNow we consider a closed orientable 3-manifold *M*. Then the tangent bundle *T**M* of *M* is always parallelizable, i.e., there always exists a global smooth trivialization (isomorphism of vector bundles) *f* : *T**M* → *M* × R3 of *T**M* (cf. and ). We call such trivialization *f* a framing of *M*.\nNow we fix a smooth framing *f* of *M*, then we always identify *T**M* with *M* × R3 and the sphere bundle *S*(*T**M*) with *M* × S2. Set *T**V*(*T**M*), *T**V*(*S*(*T**M*)) the vertical tangent bundles for the fibrations *T**M* → *M*, *S*(*T**M*) → *M* respectively. Then the above identifications induce the splittings\n \n$$\\begin{split}\n &T(TM)=f^\\*TM\\oplus f^\\*T{\\mathbb{R}}^3 =:T^H\\_f(TM)\\oplus T^V(TM),\\;\\\\ &T(S(TM))=f^\\*TM\\oplus f^\\*T\\mathbb{S}^2=:T^H\\_f(S(TM))\\oplus T^V(S(TM)).\n\\end{split}$$ \nA differential form *α* on *T**M* or *S*(*T**M*) is said to be *f*-vertical (or simply, vertical, when the framing *f* is fixed) if for all *U* ∈ *T**f**H*(*T**M*) or *T**f**H*(*S*(*T**M*)), we have\n \n*ι**U**α* = 0, \nwhere *ι**U* denotes the contraction of *U*.\nGiven an orientation *o*(*M*), via *f*, we also obtain an orientation *o*(R3) of R3. The standard unit 2-sphere S2 is boundary of the standard 3-ball, then let *o*ind(S2) be the induced orientation. At the same time, the complementary part of open 3-ball in R3 can be identified with the real blow up *B*ℓ(R3, 0) of the origin point 0 of R3. Then ∂*B*ℓ(R3, 0) = S2. We orient *B*ℓ(R3, 0) as for R3. Then\n \n*o*ind(∂*B*ℓ(R3, 0)) = − *o*ind(S2). \nIf *N*1, *N*2 are two oriented manifolds with orientations *o*(*N*1), *o*(*N*2) respectively. Then we denote by *o*(*N*1 × *N*2) the orientation of *N*1 × *N*2 given by *o*(*N*1) ∧ *o*(*N*2). This way, we orient product manifolds *M**n* and also Conf*n*(*M*), *C**k*(*M*). In particular, let *o*(*B*ℓ(*M*2, Δ)) = *o*(*C*2(*M*)) be the induced orientation.\nSince *f* also identifies *S*(*T**M*) with *M* × S2. This way, let *o*(*S*(*T**M*)) denote the orientation *o*(*M*) ∧ *o*ind(S2).\nUnder the identification ∂*C*2(*M*) ≃ *S**ν*Δ ≃ *S*(*T**M*) as explained in, we have\n \n*o*ind(∂*C*2(*M*)) = *o*(*S*(*T**M*)). \nIn the sequel where we compute the integrals on *C*2(*M*), we always fix the orientation *o*(*C*2(*M*)) induced from *o*(*M* × *M*), and we fix the induced orientation *o*ind(∂*C*2(*M*)) for ∂*C*2(*M*).\nAdjoint local system and diagonal classes\n=========================================\nThis section recalls several basic facts on local systems *E**ρ* given via adjoint representations of semi-simple Lie groups, for example, de Rham cohomology groups of such a local system, the diagonal class, the Killing form and its associated cubic trace form, and Lie structure on *E**ρ*. At last, we give an important result in Proposition [prop:3.-1] on the nonvanishing of the cohomology group *H*−2(*M*; *E**ρ* ⊗ *E**ρ*), this cohomology group plays an important role in our construction of (adapted) propagators in the next sections.\nNow we fix a connected semi-simple Lie group *G* with Lie algebra g, and let *B* : g × g → R denote the corresponding Killing form. Let *ρ* : *π*1(*M*) → *G* be a morphism of groups, composing with the adjoint action Ad : *G* → Aut(g), we get a representation of *π*1(*M*) with representation space g, which is still denoted by *ρ*. The representation (*ρ*, g) defines canonically a flat vector bundle (*E**ρ*, ∇*E**ρ*) (equivalently, a local system) on *M*: more precisely, let $\\widetilde{M}$ be the universal cover of *M* on which *π*1(*M*) acts smoothly and freely, then \n$$E\\_{\\rho}=\\pi\\_{1}(M)\\backslash \\big(\\widetilde{M}\\times\\_{\\rho} \n\\mathfrak{g}\\big),$$ \n and the flat connection ∇*E**ρ* is induced from the usual differential on $\\widetilde{M}$. In the sequel of this section, we use the terminology *flat vector bundle* for *E**ρ* in order to explain the associated de Rham–Hodge theory, and in the most part of other sections, we will call *E**ρ* the local system (associated to a representation *ρ* as above).\nde Rham cohomology and Poincaré duality\n---------------------------------------\nThe flat connection ∇*E**ρ* defines canonically a twisted differential *d* acting on Ω•(*M*; *E**ρ*), the differential forms on *M* valued in *E**ρ*. This way, we get the de Rham complex (Ω•(*M*; *E**ρ*), *d*). Let *H*•(*M*; *E**ρ*) denote the corresponding de Rham cohomology groups.\nAt the same time, viewing *E**ρ* as a local system, the singular homology groups *H*•(*X*; *E**ρ*) and singular cohomology groups *H*sing•(*M*, *E**ρ*) are well-defined, which is also defined via considering the triangulations of *M*. Of course, we have the caonical isomorphism *H*sing•(*M*; *E**ρ*) ≃ *H*•(*M*; *E**ρ*).\nNote that we always fix an orientation *o*(*M*) on *M*. For the flat vector bundle *E**ρ*, we consider its dual bundle *E**ρ*∨ : = Hom(*E**ρ*, R) on *M*, and let ∇*E**ρ*∨ denote the induced flat connection from (*E**ρ*, ∇*E**ρ*).\nThe classical Poincaré pairing (cf., ) is given as the first part of the following isomorphisms for *k* = 0, 1, 2, 3,\n \n*H**k*(*M*; *E**ρ*) ≃ *H*sing3 − *k*(*M*; *E**ρ*) ≃ *H*3 − *k*(*M*; *E**ρ*). \nThe analogous Poincaré pairing in terms of de Rham cohomology groups is given by the following nondegenerate bilinear form (also cf. )\n \n*H**k*(*M*; *E**ρ*∨) × *H*3 − *k*(*M*; *E**ρ*) → R, (*α*, *β*) ↦ ∫*M*⟨*α* ∧ *β*⟩. \nwhere ⟨ ⟩ denotes the fibrewise paring between *E**ρ*∨ and *E**ρ*.\nLet (*ρ*\\*, g\\*) be the representation of *π*1(*M*) adjoint to (*ρ*, g). Then actually, we have the identification of flat vector bundles \n$$E^\\vee\\_\\rho\\simeq \\pi\\_1(M)\\backslash (\\widetilde{M}\\times\\_{\\rho^\\*}\\mathfrak{g}^\\*).$$ \n Since *B* is non-degenerate, it defines an isomorphism \n*F* : g → g\\* \n such that if *a* ∈ g, the element *F*(*a*) ∈ g\\* is defined as follows, if *b* ∈ g, \n⟨*F*(*a*), *b*⟩ = *B*(*b*, *a*). \n By the fact that *B* is Ad(*G*)-invariant, we get that *F* is actually the isomorphism between the *π*1(*M*)-representations (*ρ*, g) and (*ρ*\\*, g\\*). Hence it induces an identification of flat vector bundle (*E**ρ*, ∇*E**ρ*) to its dual bundle *E**ρ*∨ on *M*.\nAs a consequence of the above consideration, we can reformulate the pairing in as the following nondegenerate bilinear form\n \n⟨ ⋅ , ⋅ ⟩Poinc : *H*•(*M*; *E**ρ*) × *H*3 − •(*M*; *E**ρ*) → R, (*α*, *β*) ↦ ∫*M**B*(*α* ∧ *β*), \nwhere the notation *B*(*α* ∧ *β*) means taking wedge product on differential forms and contracting sections of *E**ρ* via *B*. Since *B* is symmetric, we also conclude that the pairing ⟨ ⋅ , ⋅ ⟩Poinc is symmetric.\nNow we give more details on the Poincaré pairing ⟨ ⋅ , ⋅ ⟩Poinc for *H*0(*M*; *E**ρ*) × *H*3(*M*; *E**ρ*), which is useful in Subsection [ss5.2July]. Note that *π*1(*M*) acts on g via the representation *ρ*, set\n \ng*π*1(*M*) : = {*a* ∈ g ∣ *ρ*(*γ*)*a* = *a*, for all *γ* ∈ *π*1(*M*)}. \nThen we have the natural identification\n \n*H*0(*M*; *E**ρ*) = g*π*1(*M*), \nwhere each vector *a* ∈ g*π*1(*M*) is a constant section of *E**ρ* on *M*.\nNote that g*π*1(*M*) is a Lie subalgebra of g. We also have the following result.\nThe following two statements are equivalent:\n* For any smooth 3-form *β* on *M* such that ∫*M**β* ≠ 0, the linear map \n*H*0(*M*; *E**ρ*) = g*π*1(*M*) ∋ *a* ↦ [*β* ⊗ *a*] ∈ *H*3(*M*; *E**ρ*) \n is an isomorphism, where [*β* ⊗ *a*] denotes the de Rham cohomology class of *β* ⊗ *a*;\n* the symmetric blinear form *B* : g*π*1(*M*) × g*π*1(*M*) ∋ (*a*, *b*) ↦ *B*(*a*, *b*) ∈ R is nondegenerate.\nWhen the Lie group *G* is compact (which is always assumed to be semi-simple) or more generally, (*ρ*, *E**ρ*) is a unitary representation of *π*1(*M*), *B* can induce an *ρ*(*π*1(*M*))-invariant inner product on g so that the two statements in the above lemma always hold.\nThe diagonal class\n------------------\nFor *i* ∈ {1, 2}, we denote by *p**i* : *M* × *M* → *M* the *i*-th projection map, i.e., *p**i*(*x*1, *x*2) = *x**i*.\nLet *H*•(*M*) denote the de Rham cohomology of *M* (valued in R). Then\n \n$$H^{\\bullet}(M\\times M)=H^{\\bullet}(M)\\widehat{\\otimes} H^{\\bullet}(M).\n \\label{eq:1.4.1s}$$ \nMore precisely, if *α*, *β* ∈ *H*•(*M*), then *α* ⊗ *β* = *p*1\\*(*α*) ∧ *p*2\\*(*β*) ∈ *H*•(*M* × *M*).\nLet $\\mathfrak{i}:\\Delta\\hookrightarrow M\\times M$ denote the inclusion of the diagonal. Then the pull-back map gives the morphism\n \ni\\* : *H*•(*M* × *M*) → *H*•(Δ) ≃ *H*•(*M*), \nso that if *α*, *β* ∈ *H*•(*M*), then\n \ni\\*(*p*1\\*(*α*) ⊗ *p*2\\*(*β*)) = *α* ∧ *β*. \nNow we consider the Poincaré dual of Δ in *H*3(*M* × *M*), which we still denote by Δ, i.e., for all *β* ∈ *H*•(*M* × *M*), we have\n \n∫*M* × *M*Δ ∧ *β* = ∫*M*i\\*(*β*). \nIf {*γ**i*} is a homogenous basis of *H*•(*M*), and if {*γ**i*\\*} is the corresponding dual basis of *H*•(*M*) with respect to the Poincaré pairing, i.e.,\n \n∫*M**γ**i* ∧ *γ**j*\\* = *δ**i**j*, \nthen we have\n \nΔ = ∑*i*( − 1)deg(*γ**i*\\*)*p*1\\*(*γ**i*) ∧ *p*2\\*(*γ**i*\\*) ∈ *H*3(*M* × *M*). \nMoreover, we have the following result.\n \n∫*M*i\\*(Δ) = *χ*(*M*) = 0, \nwhere *χ*(*M*) denotes the Euler characteristic number of *M*, which is 0 since *M* is odd dimensional.\nNow we go back to the case with the flat vector bundle *E**ρ* on *M*. Set\n \n$$E\\_{\\rho} \\boxtimes E\\_{\\rho}:=p\\_1^{\\ast}E\\_{\\rho} \\otimes \np\\_2^{\\ast} E\\_{\\rho} \\rightarrow M \\times M.\n\\label{eq:1.2.7}$$ \nSince *E**ρ* is flat, so is $E\\_{\\rho} \\boxtimes E\\_{\\rho}$ and let $\\nabla^{E\\_{\\rho} \\boxtimes E\\_{\\rho}}$ denote the corresponding induced flat connection. The vector bundle $E\\_\\rho\\boxtimes E\\_\\rho$ on the diagonal Δ ⊂ *M* × *M* becomes the tensor bundle *E**ρ* ⊗ *E**ρ*, which is equipped with respect to the induced flat connection ∇*E**ρ* ⊗ *E**ρ*.\nNote that we always fix an orientation for *M*. We explain how to define a diagonal class associated with *E**ρ* as in. Recall that the Poincaré pairing ⟨ ⋅ , ⋅ ⟩Poinc given in between *H*•(*M*; *E**ρ*) × *H*3 − •(*M*; *E**ρ*) is nondegenerate. Let {*α**i*} be a homogeneous basis of *H*•(*M*; *E**ρ*), {*α**i*\\*} is the dual basis of {*α**i*} with respect to the pairing ⟨ ⋅ , ⋅ ⟩Poinc.\nThe *diagonal class* $\\widetilde{\\Delta}$ is defined as\n \n$$\\widetilde{\\Delta} = \\sum\\_{i} (-1)^{\\deg (\\alpha^{\\*}\\_i)} \\alpha\\_i \\otimes \n\\alpha\\_i^{\\ast} \\in H^{\\bullet}(M; E\\_{\\rho}) \\widehat{\\otimes} H^{\\bullet}(M; E\\_{\\rho}).\n\\label{eq:1.4.8}$$ \nIt is clear that the definition of $\\widetilde{\\Delta}$ does not depend on the choice of the homogeneous basis {*α**i*}.\nBy choosing (homogeneous) representatives of a basis of *H*•(*M*; *E**ρ*), we get the corresponding embedding\n \n*ι* : *H*•(*M*; *E**ρ*) → Ω•(*M*; *E**ρ*), \nwhich means that for *α* ∈ *H**j*(*M*; *E**ρ*), *j* = 0, ⋯, 3, we have *ι*(*α*) ∈ Ω*j*(*M*; *E**ρ*) is a closed form and *α* = [*ι*(*α*)] ∈ *H**j*(*M*; *E**ρ*). It extends naturally to\n \n$$\\iota : H^{\\bullet}(M \n\\times M; E\\_{\\rho} \\boxtimes E\\_{\\rho}) \\rightarrow \\Omega^{\\bullet}(M \n\\times M; E\\_{\\rho} \\boxtimes E\\_{\\rho}).\n \\label{eq:1.4.10}$$ \nIn the sequel, we fix *ι* once and for all.\nLet *T* denote an involution on *M* × *M* such that *T*(*x*1, *x*2) = (*x*2, *x*1). Then it lifts to the vector bundle $E\\_{\\rho}\\boxtimes E\\_{\\rho}$ via exchanging the two factors of tensors, which we still denote by *T*. By restricting to Δ, the involution *T* becomes a vector bundle automorphism of *E**ρ* ⊗ *E**ρ*. In particular, the flat connections $\\nabla^{E\\_\\rho\\boxtimes E\\_\\rho}$, ∇*E**ρ* ⊗ *E**ρ* are equivariant under these *T*-actions.\nLet *a*, *b*, *c*, *d* be sections of *E**ρ* over *M*, and let $a\\boxtimes \nb$, $c\\boxtimes d$ be two sections of $E\\_{\\rho}\\boxtimes E\\_{\\rho}$ over *M* × *M*, set\n \n$$B\\_{1,2}(a\\boxtimes b, c\\boxtimes \n d)=p\\_{1}^{\\ast}(B(a,c))p\\_{2}^{\\ast}(B(b,d)).\n \\label{eq:1.4.12}$$ \nIt extends to the differential forms valued in $E\\_{\\rho}\\boxtimes E\\_{\\rho}$ as in the integrand of.\nSet\n \n$$\\widetilde{\\Delta}\\_{12}:= \\iota(\\widetilde{\\Delta})=\\sum\\_i \n (-1)^{\\deg(\\alpha^{\\*}\\_i)} p\\_1^{\\ast} \\iota(\\alpha\\_i) \\wedge p\\_2^{\\ast} \\iota(\\alpha\\_i^{\\ast}), \n \\label{eq:1.4.11}$$ \nthen $\\widetilde{\\Delta}\\_{12}$ is a smooth 3-form on *M* × *M* valued in $E\\_{\\rho}\\boxtimes E\\_{\\rho}$.\nThe closed 3-form $\\widetilde{\\Delta}\\_{12}\\in \\Omega^3(M\\times M; E\\_\\rho\\boxtimes E\\_\\rho)$ satisfies the following properties:\n* the cohomology class $[\\widetilde{\\Delta}\\_{12}]=\\widetilde{\\Delta}\\in H^3(M\\times M; E\\_\\rho\\boxtimes E\\_\\rho)$;\n* $T^\\ast \\widetilde{\\Delta}\\_{12}= -\\widetilde{\\Delta}\\_{12}$;\n* if $\\beta\\in \nH^{\\bullet}(M\\times M; E\\_{\\rho}\\boxtimes E\\_{\\rho})$, then\n \n$$\\int\\_{M \\times M}B\\_{1,2} (\n \\widetilde{\\Delta}\\_{12}\\wedge\\iota(\\beta)) = \\int\\_{M} \\mathfrak{i}^{\\ast}B(\\iota(\\beta)).\n \\label{eq:1.4.13}$$\nWe have\n \n$$\\int\\_{\\Delta}B(\\mathfrak{i}^{\\ast}(\\widetilde{\\Delta}\\_{12}))=0.\n \\label{eq:eulerflat}$$ \nBy, and, we get\n \n$$\\int\\_{\\Delta}B(\\mathfrak{i}^{\\ast}(\\widetilde{\\Delta}\\_{12}))=\\chi(M,E\\_{\\rho}),\n \\label{eq:eulerarg}$$ \nwhere\n \n*χ*(*M*, *E**ρ*) = ∑*j*( − 1)*j*dim*H**j*(*M*, *E**ρ*), \nis the Euler characteristic number of *M* associated with the local system *E**ρ*. By the Poincaré duality, we have\n \n*χ*(*M*, *E**ρ*) = − *χ*(*M*, *E**ρ*) = 0. \nThis completes the proof of our lemma.\nAs in, after introducing the Riemannian metric on *M*, the diagonal class $\\widetilde{\\Delta}$ can be regarded as the integral kernel of the harmonic projection *π**h* : Ω•(*M*; *E**ρ*) → *H*•(*M*; *E**ρ*), that is,\n \n$$\\begin{split} \n\\pi\\_h(\\beta) &= p\\_{1\\ast}(B\\_2(\\widetilde{\\Delta}\\_{12} \\wedge p\\_2^{\\ast} \\iota(\\beta)))\\\\\n&= \\int\\_{2} B\\_2(\\widetilde{\\Delta}\\_{12}\\wedge p\\_2^{\\ast} \\iota(\\beta))\n \\end{split}\n \\label{eq:1.4.14}$$ \nCharacter variety of *π*1(*M*) and *H*1(*M*; *E**ρ*)\n----------------------------------------------------\nThis subsection recalls some basic facts on the *G*-character variety of the fundamental group *π*1(*M*) of a closed 3-manifold *M* and its relation with the first cohomology group *H*1(*M*; *E**ρ*). Fore more details, see,, and.\nSince *M* is closed, its fundamental group *π*1(*M*) admits a finite presentation given as *π*1(*M*) = ⟨*x*1, …, *x**k* ∣ *r*1, …, *r**m*⟩. Then, the **G*-representation space* *R*(*M*, *G*) of *π*1(*M*) is defined as $R(M, G) = {\\mathop{\\mathrm{Hom}}\\nolimits}(\\pi\\_1(M), G)$. Note that there is an embedding $R(M,G) \\hookrightarrow G^k$ given by sending each representation *φ* to the *k*-tuple (*φ*(*x*1), …, *φ*(*x**k*)) ∈ *G**k*. Thus, *R*(*M*, *G*) can be viewed as a real algebraic set (or algebraic variety if *G* is complex) and it is independent of chosen presentations. Since *G* acts on $R(M, G) = {\\mathop{\\mathrm{Hom}}\\nolimits}(\\pi\\_1(M), G)$ by diagonal conjugation, the **G*-character variety* X*G*(*M*) of *π*1(*M*) is defined as categorical quotient of *R*(*M*, *G*) by the *G*-conjugate action.\nBy considering the Zariski tangent space to $R(M, G) = {\\mathop{\\mathrm{Hom}}\\nolimits}(\\pi\\_1(M), G)$ at *φ* ∈ *R*(*M*, *G*), it turns out that there is a canonical identification\n \n*T**φ**R*(*M*, *G*) = *Z**φ*1(*π*1(*M*), g) \nwhere *Z**φ*1(*π*1(*M*), g) is the space of 1-cocycles, i.e., maps *z* : *π*1(*M*) → g satisfying the 1-cocycle condition *z*(*γ*1*γ*2) = *z*(*γ*1) + *γ*1 ⋅ *z*(*γ*2) for any *γ*1, *γ*2 ∈ *π*1(*M*). Here, *π*1(*M*) action on g is given by $\\gamma \\cdot v := (({\\mathop{\\mathrm{Ad}}\\nolimits}\\circ \\varphi)(\\gamma))(v)= {\\mathop{\\mathrm{Ad}}\\nolimits}(\\varphi(\\gamma))(v)$ for any *γ* ∈ *π*1(*M*) and *v* ∈ g. Moreover, it gives rise to an isomorphism between the tangent space of the character variety X*G*(*M*) and the first cohomology group of *π*1(*M*) with coefficient in the *π*1(*M*)-module g via ${\\mathop{\\mathrm{Ad}}\\nolimits}\\circ \\varphi: \\pi\\_1(M) \\rightarrow \\mathfrak{g}$\n \n*T*[*φ*]X*G*(*M*) = *H**φ*1(*π*1(*M*), g) = *H*1(*M*; *E**ρ*) \nfor $\\rho = {\\mathop{\\mathrm{Ad}}\\nolimits}\\circ \\varphi : \\pi\\_1(M) \\rightarrow {\\mathop{\\mathrm{Aut}}\\nolimits}(\\mathfrak{g})$. For the last isomorphism, see for example.\nTherefore, for $\\rho = {\\mathop{\\mathrm{Ad}}\\nolimits}\\circ \\varphi$, if the class [*φ*] is isolated point in the *G*-character variety X*G*(*M*) of *π*1(*M*), then *H*1(*M*, *E**ρ*) = 0 (but its converse does not always hold).\nAn element *φ* ∈ *R*(*M*, *G*) is called *irreducible* when its centralizer coincides with the center of *G* (cf. ). Recall that, when *G* is connected, the kernel of the adjoint action ${\\mathop{\\mathrm{Ad}}\\nolimits}: G \\rightarrow {\\mathop{\\mathrm{Aut}}\\nolimits}(\\mathfrak{g})$ coincides with the center of *G*. Therefore, one has *H*0(*M*; *E**ρ*) = g*π*1(*M*) = 0 if *G* is connected and *φ* is irreducible.\nWe end this subsection by giving an important class of examples that satisfy acyclic condition *H*•(*M*; *E**ρ*) = 0 for the convenience of the readers.\n(Fintushel–Stern (, see also )) Let *n* ≥ 3 be a fixed integer and *a*1, …, *a**n* be pairwise relatively prime integers with *a**i* ≥ 2. Let Σ = Σ(*a*1, …, *a**n*) be the Seifert homology sphere determined by (*a*1, …, *a**n*) (cf. ), which is the closed orientable 3-manifold to be considered, and let $\\varphi : \\pi\\_1(\\Sigma) \\rightarrow {\\mathop{\\mathrm{SU}}\\nolimits}(2)$ be an irreducible representation. Recall that *π*1(Σ) admits a finite presentation as\n \n*π*1(Σ) = ⟨*x*1, …, *x**n*, *h* ∣ [*h*, *x**i*] = 1, *x**i**a**i**h**b**i* = 1, *x*1⋯*x**n* = 1⟩ \nwhere *b*1, …, *b**n* are integers satisfying the equation\n \n$$a\\_1 \\cdots a\\_n \\cdot \\sum\\_{i=1}^n \\frac{b\\_i}{a\\_i} = 1.$$ \nSuppose that, for an integer *m* ≥ 3, *φ*(*x**k*) ≠ ± 1 for *k* = 1, …, *m* and *φ*(*x**k*) = ± 1 for *k* = *m* + 1, …, *n*. Then, it turns out that *H**φ*1(*π*1(Σ), su(2)) = *H*1(Σ; *E**ρ*) = R2*m* − 6 with $\\rho = {\\mathop{\\mathrm{Ad}}\\nolimits}\\circ \\varphi.$ In particular, when *m* = 3, one obtains *H*1(Σ; *E**ρ*) = 0. Since *φ* is irreducible, we have *H*0(Σ; *E**ρ*) = 0. Therefore, using Poincaré duality, we conclude *H*•(Σ; *E**ρ*) = 0 and *ρ* with *m* = 3 gives an acyclic local system on Σ.\nKilling form, cubic trace form\n------------------------------\nRecall that the Killing form *B* ∈ g\\* ⊗ g\\* is a non-degenerate bilinear form, we have the corresponding Casimir element **1** ∈ g ⊗ g. Let *e*1, …, *e*dimg be a basis of g, and let *e*1\\*, …, *e*dimg\\* ∈ g be dual basis of {*e**i*} with respect to *B*, i.e., *B*(*e**i*, *e**j*\\*) = *δ**i**j*. Then **1** can be explicitly written as\n \n**1** = ∑*i* = 1dimg*e**i* ⊗ *e**i*\\*. \nSince *B* is invariant under the adjoint action of *π*1(*M*), it induces a fiberwise non-degenerate bilinear form on the vector bundle *E**ρ* → *M*. By abuse of notation, we use the same *B* to denote it, i.e., we have\n \n$$B : E\\_{\\rho} \\otimes E\\_{\\rho} \\rightarrow \\underline{\\mathbb{R}},\n \\label{eq:1.3.2}$$ \nwhere $\\underline{\\mathbb{R}}$ stands for the trivial local system on *M*.\nMoreover, the element **1** is fixed by the diagonal action of *π*1(*M*) on g ⊗ g. This way, we can view **1** as a smooth section of *E**ρ* ⊗ *E**ρ* on *M*. Moreover, we get a well-defined map\n \n$$I : \\underline{\\mathbb{R}} \\rightarrow E\\_{\\rho} \\otimes E\\_{\\rho}, \\quad 1 \\mapsto \\mathbf{1}.\n \\label{eq:1.3.3}$$ \nThe following result is clear by definition.\n[lm:1.1] The section **1** is a flat section.\nNow we introduce the cubic trace form. If *a*, *b*, *c* ∈ g, set\n \nTr[*a* ⊗ *b* ⊗ *c*] = *B*([*a*, *b*], *c*). \nThen Tr ∈ Λ3(g\\*), which is Ad(*G*)-invariant. Then it extends to Λ3(*E**ρ*\\*), which is flat with respect to ∇*E**ρ*.\nLie bracket on *E**ρ*\n---------------------\nWe set L : = [ ⋅ , ⋅ ] : g ⊗ g → g, *a* ⊗ *b* ↦ [*a*, *b*]. Let h ⊂ g ⊗ g be the kernel space of L. Since g is semisimple, then L is surjective. Then\n \ndimRh = (dimRg)2 − dimRg. \nIn particular, the symmetric tensor space *S*2g is a subspace of h.\nRecall that *π*1(*M*) acts on g ⊗ g by the diagonal action via *ρ*. The Lie bracket operator L is equivariant with respect to the actions of *π*1(*M*). Then *π*1(*M*)-action preserves the subspace h. Set the vector bundle\n \n$$H\\_\\rho=\\pi\\_1(M)\\backslash(\\widetilde{M}\\times\\_\\rho \\mathfrak{h}).$$ \nIt is equipped with the canonically induced flat connection ∇*H**ρ* and is a subbundle of *E**ρ* ⊗ *E**ρ*.\nMoreover, the Lie bracket operator L defines canonically a morphism of vector bundles on *M*,\n \nL : *E**ρ* ⊗ *E**ρ* → *E**ρ*. \nBy construction, it preserves the flat connections, i.e., when acting on smooth sections, ∇*E**ρ* ∘ L = L ∘ ∇*E**ρ* ⊗ *E**ρ*. We have the following properties:\n* If *s**i*, *i* = 1, 2, 3 are smooth sections of *E**ρ* on *M*, then\n \n$$\\begin{split}\n &\\mathfrak{L}(s\\_{1}\\otimes s\\_{2})=-\\mathfrak{L}(s\\_{2}\\otimes s\\_{1}),\\\\\n &\\mathfrak{L}(\\mathfrak{L}(s\\_{1}\\otimes s\\_{2})\\otimes s\\_{3})+\\mathfrak{L}(\\mathfrak{L}(s\\_{2}\\otimes \n s\\_{3})\\otimes s\\_{1})+\\mathfrak{L}(\\mathfrak{L}(s\\_{3}\\otimes s\\_{1})\\otimes \n s\\_{2})=0.\n \\end{split}\n \\label{eq:1.5.2}$$\n* We also have\n \nTr[*s*1 ⊗ *s*2 ⊗ *s*3] = *B*(L(*s*1 ⊗ *s*2) ⊗ *s*3) ∈ *C*∞(*M*).\nWe get the following identification of vector bundles\n \n*H**ρ* = ker(L : *E**ρ* ⊗ *E**ρ* → *E**ρ*), \nwhere ∇*H**ρ* is exactly the one inherited from the flat connection ∇*E**ρ* ⊗ *E**ρ*.\nWe also extend it on Ω•(*M*, *E**ρ*) such that if *α*, *β* ∈ Ω•(*M*), *s*1, *s*2 ∈ *C*∞(*M*, *E**ρ*),\n \nL(*α**s*1 ⊗ *β**s*2) = *α* ∧ *β*L(*s*1 ⊗ *s*2). \nThen L induces the morphism of de Rham cohomology groups,\n \nL : *H*•(*M*; *E**ρ* ⊗ *E**ρ*) → *H*•(*M*; *E**ρ*). \nBy our construction, we have the short exact sequence of flat vector bundles: \n$$0\\rightarrow H\\_{\\rho}\\hookrightarrow \nE\\_{\\rho}\\otimes E\\_{\\rho}\\rightarrow E\\_{\\rho}\\rightarrow 0.$$ \n Then we obtain the following long exact sequence of cohomology groups,\n \n$$\\begin{split}\n &\\cdots\\rightarrow H^{0}(M; E\\_{\\rho})\\rightarrow \n H^{1}(M; H\\_{\\rho})\\rightarrow \n H^{1}(M; E\\_{\\rho}\\otimes E\\_{\\rho}) \\overset{\\mathfrak{L}}{\\longrightarrow}\n H^{1}(M; E\\_{\\rho})\\\\\n &\\qquad\\qquad\\rightarrow \n H^{2}(M; H\\_{\\rho}) \\rightarrow \n H^{2}(M; E\\_{\\rho}\\otimes E\\_{\\rho}) \\overset{\\mathfrak{L}}{\\longrightarrow}\n H^{2}(M; E\\_{\\rho})\\rightarrow \\cdots \n \\end{split}\n \\label{eq:1.5.6}$$ \nNote that *T* acts on *M* as identity but exchanges the factors of the tensor *E**ρ* ⊗ *E**ρ*. It induces the involutions *T*\\* acting on the de Rham complexes, and hence on the cohomology groups *H*•(*M*; *H**ρ*), *H*•(*M*; *E**ρ* ⊗ *E**ρ*). Let Ω±•(*M*; *H**ρ*), Ω±•(*M*; *E**ρ* ⊗ *E**ρ*), *H*±•(*M*; *H**ρ*), *H*±•(*M*; *E**ρ* ⊗ *E**ρ*) denote the eigenspaces of *T*\\* corresponding to the eigenvalues ± 1. One can verify directly that for each class [*α*] ∈ *H*±•(*M*; ⋯), it always admits a representative *α* ∈ Ω±•(*M*; ⋯). In particular, we have the splitting of the de Rham complexes\n \n(Ω•(*M*; ⋯), *d*) = (Ω+•(*M*; ⋯), *d*) ⊕ (Ω−•(*M*; ⋯), *d*), \nwhich also corresponds to the splitting of the associated cohomology groups\n \n*H*•(*M*; ⋯) = *H*+•(*M*; ⋯) ⊕ *H*−•(*M*; ⋯). \n[ex:3.5.1] We consider the flat section **1** of *E**ρ* ⊗ *E**ρ* over *M* defined in previous subsection, i.e., **1** ∈ *H*0(*M*; *E**ρ* ⊗ *E**ρ*). A straightforward computation shows that L(**1**) = 0, hence we have **1** ∈ *H*0(*M*; *H**ρ*). Since *B* is a symmetric bilinear form, we also conclude *T*\\***1** = **1**, i.e., **1** ∈ *H*+0(*M*, *H**ρ*).\n[prop:3.5.2paris] We have the following isomorphism\n \n*H*+•(*M*; *H**ρ*) ≃ *H*+•(*M*; *E**ρ* ⊗ *E**ρ*). \nWe also have the long exact sequence as follows\n \n$$\\begin{split}\n &\\cdots\\rightarrow H^{0}(M; E\\_{\\rho})\\rightarrow \n H^{1}\\_{-}(M; H\\_{\\rho})\\rightarrow \n H^{1}\\_{-}(M; E\\_{\\rho}\\otimes E\\_{\\rho}) \\overset{\\mathfrak{L}}{\\longrightarrow}\n H^{1}(M; E\\_{\\rho})\\\\\n &\\qquad\\qquad\\rightarrow \n H^{2}\\_{-}(M; H\\_{\\rho}) \\rightarrow \n H^{2}\\_{-}(M; E\\_{\\rho}\\otimes E\\_{\\rho}) \\overset{\\mathfrak{L}}{\\longrightarrow} \n H^{2}(M; E\\_{\\rho})\\rightarrow \\cdots \n \\end{split}\n \\label{eq:3.5.10paris}$$ \nWe split g ⊗ g as the anti-symmetric part and symmetric part, then we get\n \ng ⊗ g = Λ2g ⊕ *S*2g. \nThis splitting corresponds to the eigenspace decomposition of the involution *T* on g ⊗ g. Then we get an exact sequence of vector spaces\n \n$$0\\rightarrow \\Lambda^2\\mathfrak{g}\\cap \\mathfrak{h}\\rightarrow \\Lambda^2\\mathfrak{g}\\overset{\\mathfrak{L}}{\\longrightarrow}\\mathfrak{g}\\rightarrow 0.$$ \nThis exact sequence is compatible with the action *π*1(*M*), which lifts to the short exact sequence of associated local systems. Hence we can get the long exact sequence. Then combining it with, we get.\nIf *E**ρ* is acyclic, then by the above long exact sequence, we get a canonical isomorphism\n \n*H*±•(*M*; *H**ρ*) ≃ *H*±•(*M*; *E**ρ* ⊗ *E**ρ*). \nAs a consequence, for each cohomological class [*ξ*] ∈ *H*±•(*M*; *E**ρ* ⊗ *E**ρ*), there exists a closed form *ξ*0 ∈ Ω±•(*M*; *H**ρ*), such that [*ξ*0] = [*ξ*] and\n \nL(*ξ*0) = 0. \nBy Example [ex:3.5.1], we see that **1** ∈ *H*+0(*M*; *E**ρ* ⊗ *E**ρ*), which means that *E**ρ* ⊗ *E**ρ* can never be acyclic.\nThe following proposition can be viewed as an extension of.\n[prop:3.5.2] If *G* is a real 3-dimensional simple Lie group, then we have\n \n*H*−•(*M*; *H**ρ*) = 0, *H*−•(*M*; *E**ρ* ⊗ *E**ρ*) ≃ *H*•(*M*; *E**ρ*). \nIn particular, if in addition *E**ρ* is acyclic, then \n*H*−•(*M*; *E**ρ* ⊗ *E**ρ*) = 0. \nWe need to point out that if *G* is semisimple and real 3-dimensional, then it has to be a simple Lie group. In fact, for such linear Lie group *G*, if *G* is compact, then *G* = SU(2) or SO(3); if *G* is noncompact, then *G* = SL2(R) or SO(2, 1).\n[Proof of Proposition [prop:3.5.2]] When g is simple with dimRg = 3, we conclude directly h = *S*2g. Then *T*\\* acts on *H*•(*M*; *H**ρ*) as identity, we get\n \n*H*+•(*M*; *H**ρ*) = *H*•(*M*; *H**ρ*), *H*−•(*M*; *H**ρ*) = 0. \nThen this proposition follows from Proposition [prop:3.5.2paris].\nAt last, we give an example of a pair (*M*, *ρ*) such that *H*•(*M*; *E**ρ*) = 0, but *H*−1(*M*; *E**ρ* ⊗ *E**ρ*) = *H*−2(*M*; *E**ρ* ⊗ *E**ρ*) ≠ 0 when $G={\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})\\times {\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})$.\n[prop:3.-1] Let *M* be an oriented closed hyperbolic 3-manifold that contains a totally geodesic surface. Let \n$$\\rho:\\pi\\_1(M) \\overset{\\widetilde{\\mathrm{hol}}}{ \\xrightarrow{\\hspace\\*{0.5cm}}} {\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C}) \\overset{{\\mathop{\\mathrm{Id}}\\nolimits}\\times \\overline{\\,{\\mathop{\\mathrm{Id}}\\nolimits}\\,}}{ \\xrightarrow{\\hspace\\*{1cm}}}{\\mathop{\\mathrm{SL}}\\nolimits}(2,\\mathbb{C}) \\times {\\mathop{\\mathrm{SL}}\\nolimits}(2,\\mathbb{C}) \\overset{{\\mathop{\\mathrm{Ad}}\\nolimits}}{\\longrightarrow} {\\mathop{\\mathrm{Aut}}\\nolimits}(\\mathfrak{sl}\\_2(\\mathbb{C}) \\oplus \\mathfrak{sl}\\_2(\\mathbb{C}))$$ \n where $\\pi\\_1(M) \\overset{\\widetilde{\\mathrm{hol}}}{\\longrightarrow} {\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})$ is a lift of the holonomy representation $\\mathrm{hol}: \\pi\\_1(M) \\rightarrow {\\mathop{\\mathrm{PSL}}\\nolimits}(2, \\mathbb{C})$ corresponding to the complete hyperbolic structure of *M* and $\\overline{\\,{\\mathop{\\mathrm{Id}}\\nolimits}\\,}: {\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C}) \\rightarrow {\\mathop{\\mathrm{SL}}\\nolimits}(2,\\mathbb{C})$ denotes the complex conjugation of matrix. Then, we have\n \n*H*•(*M*; *E**ρ*) = 0, *H*−1(*M*; *H**ρ*) ≃ *H*1(*M*; Λ2*E**ρ*) ≠ 0. \nTo prove the statement we use several facts about hyperbolic 3-manifolds, for example, summarized in Porti’s article. Let *V*2, 0 denote the space of two variable degree 2 homogeneous polynomials with coefficients in C. Then, ${\\mathop{\\mathrm{SL}}\\nolimits}(2,\\mathbb{C})$ acts on *V*2, 0 by (*A*, *P*) ↦ *P* ∘ *A**t* for $A \\in {\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})$ and *P* ∈ *V*2, 0, where *A**t* denotes the transposition of *A*. We set $V\\_{0,2}:= \\overline{V\\_{2,0}}$ the complex conjugate representation to *V*2, 0. Then, as is well known *V*2, 0, *V*0, 2 and *V*2, 2 : = *V*2, 0 ⊗ *V*0, 2 are irreducible representations of ${\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})$.\nLet $\\mathfrak{sl}\\_2(\\mathbb{C})\\_{{\\mathop{\\mathrm{Ad}}\\nolimits}}$ and $\\mathfrak{sl}\\_2( \\mathbb{C})\\_{\\overline{{\\mathop{\\mathrm{Ad}}\\nolimits}}}$ denote the sl2(C) as ${\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})$-modules via adjoint representation ${\\mathop{\\mathrm{Ad}}\\nolimits}:{\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})\\rightarrow {\\mathop{\\mathrm{Aut}}\\nolimits}(\\mathfrak{sl}\\_2(\\mathbb{C}))$ and its complex conjugate $\\overline{{\\mathop{\\mathrm{Ad}}\\nolimits}}$ respectively. Then, as ${\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})$-modules we have $\\mathfrak{sl}\\_2(\\mathbb{C})\\_{{\\mathop{\\mathrm{Ad}}\\nolimits}} \\simeq V\\_{2,0}$ and $\\mathfrak{sl}\\_2(\\mathbb{C})\\_{\\overline{{\\mathop{\\mathrm{Ad}}\\nolimits}}} \\simeq V\\_{0,2}$. Hence, *H*•(*M*; *E**ρ*) = *H*•(*M*; *V*2, 0) ⊕ *H*•(*M*; *V*0, 2) = 0 by Raghunathan vanishing theorem (, ), irreducibility of representations *V*2, 0 and *V*0, 2, and Poincaré duality. Here, by abuse of notation, we denote the local systems associated with $\\pi\\_1(M) \\overset{\\widetilde{\\mathrm{hol}}}{ \\rightarrow} {\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C}) \\rightarrow {\\mathop{\\mathrm{GL}}\\nolimits}(V)$ for *V* = *V*2, 0, *V*0, 2 by the same symbols *V*2, 0 and *V*0, 2 respectively. We have isomorphisms as ${\\mathop{\\mathrm{SL}}\\nolimits}(2, \\mathbb{C})$-modules\n \n$$\\begin{split}\n & \\Lambda^2(\\mathfrak{sl}\\_2(\\mathbb{C})\\_{{\\mathop{\\mathrm{Ad}}\\nolimits}} \\oplus \\mathfrak{sl}\\_2(\\mathbb{C})\\_{\\overline{{\\mathop{\\mathrm{Ad}}\\nolimits}}}) \\\\\n & \\simeq \\Lambda^2( V\\_{2,0} \\oplus V\\_{0,2}) \\\\\n & \\simeq \\Lambda^2(V\\_{2,0}) \\oplus (V\\_{2,0} \\otimes V\\_{0,2}) \\oplus \\Lambda^2 (V\\_{0,2})\\\\\n & \\simeq V\\_{2,0} \\oplus V\\_{2,2} \\oplus V\\_{0,2},\n\\end{split}$$ \nwhere we refer, for example, for the last isomorphism. Again by Raghunathan vanishing theorem, we get\n \n*H*1(*M*; Λ2*E**ρ*) ≃ *H*1(*M*; *V*2, 2). \nThen, by Millson’s theorem (, ), we conclude *H*1(*M*; *V*2, 2) ≠ 0. The assertion is proved.\nIt is known that there are infinitely many hyperbolic rational homology spheres containing closed embedded totally geodesic surfaces. Therefore, by combining this fact with Proposition [prop:3.-1], one sees that there are infinitely many examples of a pair (*M*, *ρ*) which satisfies the condition *H*•(*M*; *E**ρ*) = 0 and *H*−1(*M*; *E**ρ* ⊗ *E**ρ*) = *H*−2(*M*; *E**ρ* ⊗ *E**ρ*) ≠ 0.\nConstruction of the propagator on *C*2(*M*)\n===========================================\nThis section describes a general recipe to construct propagators, from which one can define configuration space integrals, for non-acyclic local systems inspired by,, and. Then, we introduce a class of propagators, called adapted propagators, for acyclic local systems. They have a distinguished feature that is crucial for our results in subsequent sections.\nRecall that we denote by *p**i* : *M* × *M* → *M* the *i*-th projection map, i.e., *p**i*(*x*1, *x*2) = *x**i*. In the sequel, we also denote the *i*-th projection map by the same notation *p**i* : *C*2(*M*) → *M*, so that the blow-down map *q* : *C*2(*M*) → *M* × *M* can be written as *q* = (*p*1, *p*2). Moreover, for *n* ≥ 2, the projection (*i* ≠ *j*)\n \n*p**i**j* : Conf*n*(*M*) ∋ (*x*1, …, *x**n*) ↦ (*x**i*, *x**j*) ∈ Conf2(*M*) \ninduces a smooth map of manifolds with corners\n \n*C**n*(*M*) → *C*2(*M*) \nwhich is still denoted by *p**i**j*. By Proposition [prop:subme], *p**i**j* is a submersion for manifolds with corners.\nWe define the pull-back vector bundle $F\\_{\\rho}:= q^{\\ast}(E\\_{\\rho} \n\\boxtimes E\\_{\\rho}) \\rightarrow C\\_2(M)$. Note that *F**ρ*∣∂*C*2(*M*) is just the pull-back of *E**ρ* ⊗ *E**ρ* → *M* by the projection *q*∂ : ∂*C*2(*M*) → Δ ≃ *M*, which we still denote by *E**ρ* ⊗ *E**ρ*. Moreover, we have the corresponding induced flat connection ∇*F**ρ* on *F**ρ*.\nRegular forms\n-------------\nWe recall the notion of regular forms on compactified configuration space, introduced in. In some sense, regular forms encode the non-singular part of a form on *M**n* which may diverge along the diagonals but has a smooth extension to *C**n*(*M*).\nLet *q* : *C**n*(*M*) → *M**n* be the blow-down map. Then, a form in $\\Omega^{\\bullet}(C\\_n(M); q^{\\ast}(E\\_{\\rho}^{\\boxtimes n}))$ is called *regular* if it is the pullback *q*\\*(*ξ*) of a form $\\xi \\in \\Omega^{\\bullet}(M^n;E\\_{\\rho}^{\\boxtimes n})$ via *q*. We denote by $\\Omega^{\\bullet}\\_{\\mathrm{reg}}(C\\_n(M);q^{\\ast}(E\\_{\\rho}^{\\boxtimes n}))$ the space of $q^{\\ast}(E\\_{\\rho}^{\\boxtimes n})$-valued regular forms on *C**n*(*M*).\nFor two differential forms *α*, *β* ∈ Ω•(*C*2(*M*); *F**ρ*) one of which is regular, their convolution is defined by *α* \\* *β* : = *p*13, \\*(*B*2(*p*12\\**α* ∧ *p*23\\**β*)). Then, the convolution \\* defines bilinear maps as,\n \n$$\\begin{split}\n &\\ast: \\Omega^{\\bullet}(C\\_2(M); F\\_{\\rho})\\times \\Omega^{\\bullet}\\_{\\mathrm{reg}}(C\\_2(M); F\\_{\\rho}) \\rightarrow \\Omega^{\\bullet}\\_{\\mathrm{reg}}(C\\_2(M); F\\_{\\rho})\\\\\n &\\ast: \\Omega^{\\bullet}\\_{\\mathrm{reg}}(C\\_2(M); F\\_{\\rho}) \\times \\Omega^{\\bullet}(C\\_2(M); F\\_{\\rho}) \\rightarrow \\Omega^{\\bullet}\\_{\\mathrm{reg}}(C\\_2(M); F\\_{\\rho}).\n \\end{split}$$ \nFor example, for *α* = *q*\\**α*ʹ ∈ Ωreg•(*C*2(*M*); *F**ρ*) and *β* ∈ Ω•(*C*2(*M*); *F**ρ*), their convolution is given by *α* \\* *β* = *q*\\**γ* where *γ* = (pr1 × *p*2)\\**B*2((pr1 × *p*1)\\**α*ʹ ∧ pr2\\**β*), pr1 : *M* × *C*2(*M*) → *M* and pr2 : *M* × *C*2(*M*) → *C*2(*M*). Note that here we essentially use the fibration structure of pr1 × *p*2 : *M* × *C*2(*M*) → *M* × *M* to define this convolution \\* , where the condition of regular forms guarantees the well-definedness.\nAn element in *H*−2(∂*C*2(*M*); *E**ρ* ⊗ *E**ρ*)\n------------------------------------------------\nThe involution *T* of *M* × *M* → *M* × *M* given by (*x*1, *x*2) ↦ (*x*2, *x*1) extends to an involution on *C*2(*M*), which preserves the boundary ∂*C*2(*M*). It also lifts to the bundle *F**ρ* by exchanging the factors of the tensor product. Let Ω±•(*C*2(*M*); *F**ρ*) (resp. Ω±•(∂*C*2(*M*); *E**ρ* ⊗ *E**ρ*)) denote the ( ± 1)-eigenspaces of the action of *T*\\*, and we also use similar convention for the cohomology groups.\nWe consider the oriented unit sphere S2 in R3. Let *T*S2 be the involution on S2 given by *T*S2(*v*) = − *v*, *v* ∈ S2. Let *η* denote a smooth normalized volume form on the unit sphere S2 such that *T*S2\\**η* = − *η*. If *η*ʹ is another such volume form, then there exists a 1-form *γ* on S2 with *T*S2\\**γ* = − *γ* such that \n*η* − *η*ʹ = *d**γ*. \n Consider the obvious projection *π* : *M* × S2 → S2, then *π*\\**η* is a closed 2-form on *M* × S2.\nNote that the sphere normal bundle *S**ν*Δ is identified with the sphere tangent bundle *S*(*T**M*) by\n \n$$S\\nu\\_{\\Delta} \\overset{\\simeq}{\\rightarrow} S(TM), \\quad ((x,x), \n(v,-v)) \\mapsto (x, v).\n\\label{eq:2.1.1}$$ \nThe involution *T* on *C*2(*M*) restricting to the boundary corresponds to the involution on *S*(*T**M*): (*x*, *v*) ↦ (*x*, − *v*). We always use *T* to denote all these involution operators.\nThe given framing *f* of *M* induces a canonical identification *M* × S2 ≃ *S*(*T**M*) ≃ ∂*C*2(*M*). This way, we view *π*\\**η* as a closed 2-form on ∂*C*2(*M*). In particular, we have *T*\\*(*π*\\**η*) = *π*\\*(*T*S2\\**η*) = − *π*\\**η*. With the above consideration, we have the following natural identification of vector spaces\n \n$$H^2\\_{-}(\\partial C\\_2(M); E\\_\\rho\\otimes E\\_\\rho)\\simeq H^2\\_{-}(M; E\\_\\rho\\otimes E\\_\\rho)\\oplus\\big(H^2(\\mathbb{S}^2;{\\mathbb{R}})\\otimes H^0\\_{+}(M; E\\_\\rho\\otimes E\\_\\rho)\\big),\n \\label{eq:4.1.2paris}$$ \nwhere *H*2(S2; R) = R[*η*] is 1-dimensional, and the direct sum depends on the given framing *f*.\nRecall that the element **1** is defined in, it is also regarded as a flat section of *E**ρ* ⊗ *E**ρ* over *M* or *S*(*T**M*), hence on ∂*C*2(*M*). We define a 2-form on Ω2(∂*C*2(*M*); *E**ρ* ⊗ *E**ρ*) by\n \n*I*(*η*) = *π*\\**η* ⊗ **1**, \nwhere the notation *I*( ⋅ ) is compatible with the definition given in, and the form *π*\\**η* should be viewed as the 2-form on ∂*C*2(*M*) given via the pull-back of *f*, sometimes we also denote it by *f*\\**η* to emphasize the role of the framing *f*.\nThe following lemma is an analog of.\n[lm:2.1] The 2-form *I*(*η*) ∈ Ω2(∂*C*2(*M*); *E**ρ* ⊗ *E**ρ*) satisfies the following properties:\n1. *I*(*η*) is a closed form,\n2. *I*(*η*) is a fiberwise tensor product of volume form and **1**, i.e., its (*E**ρ* ⊗ *E**ρ*-valued) fiber integration is **1**,\n3. *I*(*η*) is antisymmetric under the action of *T*\\*, i.e., *T*\\*(*I*(*η*)) = − *I*(*η*).\nThe first property follows from the closeness of *η* and Lemma [lm:1.1]. The second and third properties follow from the properties of *η* and the fact that *T*\\* acts trivially on **1**.\nThe following result was already implied in \n[lem:4.1.2paris] If *f*ʹ is another smooth framing of *M* which induces the same orientation as *f* does, then by taking an oriented normalized volume form *η*ʹ on S2, the corresponding closed 2-form *I*(*η*ʹ) lies in the same de Rham cohomology class [*I*(*η*)] as of *I*(*η*) in *H*−2(∂*C*2(*M*); *E**ρ* ⊗ *E**ρ*).\nNote that *H*−2(∂*C*2(*M*); R) ≃ *H*−2(*M* × S2; R) ≃ *H*2(S2; R) is 1-dimensional vector space spanned by [*η*]. Hence [*f*\\**η*] = [(*f*ʹ)\\**η*ʹ] ∈ *H*−2(∂*C*2(*M*); R). Then there exists *β*ʹ ∈ Ω−1(∂*C*2(*M*); R) such that\n \n*f*\\**η* − (*f*ʹ)\\**η*ʹ = *d**β*ʹ. \nAs a consequence, we conclude the identity in Ω−2(∂*C*2(*M*); *E**ρ* ⊗ *E**ρ*),\n \n*I*(*η*) − *I*(*η*ʹ) = (*d**β*ʹ) ⊗ **1** = *d*(*β*ʹ ⊗ **1**). \nThis way, we conclude this lemma.\nIn the above lemma, the framing *f*ʹ is not necessary to be homotopic to *f*. When *f*ʹ is homotopic to *f*, the form *β*ʹ in can be constructed more explicitly as follows.\n[lm:4.1.3July] Fix an oriented normalized volume form *η* on S2. If *f*ʹ is another smooth framing of *M* which is homotopic to *f*, let *I*ʹ(*η*) be the closed 2-form in Ω−2(∂*C*2(*M*); *E**ρ* ⊗ *E**ρ*) defined by *f*ʹ. Then there is a *f*-vertical 1-form *β*ʹ ∈ Ω−1(∂*C*2(*M*); R) such that\n \n*I*ʹ(*η*) − *I*(*η*) = *d*(*β*ʹ ⊗ **1**). \nNote that in this case, *f*ʹ ∘ *f*− 1 is connected to the identity section by a smooth path in C∞(*M*, Diff(S2)). Let *ψ*⋅ : [0, 1] ∋ *t* ↦ *ψ**t* ∈ C∞(*M*, Diff(S2)) denote such a path with *ψ*0(*x*) = IdS2 and *ψ*1(*x*) = (*f*ʹ ∘ *f*− 1)*x*. In particular, we view *ψ**t* as a diffeomorphism of *M* × S2. Let Γ(S2) denote the space of smooth"},"data_id":{"kind":"string","value":"arxiv_0182771"}}},{"rowIdx":45693,"cells":{"content":{"kind":"string","value":"On the analytic systole of Riemannian surfaces of finite type\n=============================================================\n[1](#fn1)\nIn we introduced, for a Riemannian surface *S*, the quantity Λ(*S*) : = inf*F**λ*0(*F*), where *λ*0(*F*) denotes the first Dirichlet eigenvalue of *F* and the infimum is taken over all compact subsurfaces *F* of *S* with smooth boundary and abelian fundamental group. A result of Brooks implies Λ(*S*) ≥ *λ*0(*S̃*), the bottom of the spectrum of the universal cover *S̃*. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate Λ(*S*) with the systole, improving the main result of.\nIntroduction\n============\nSmall eigenvalues of Riemannian surfaces, in particular of hyperbolic surfaces, have been of interest in different mathematical fields for a long time. Buser and Schmutz conjectured that a hyperbolic metric on the closed surface *S* = *S**g* of genus *g* ≥ 2 has at most 2*g* − 2 eigenvalues below 1/4. In, Otal and Rosas proved a generalized version of this conjecture. They showed that a real analytic Riemannian metric on *S**g* with negative curvature has at most 2*g* − 2 eigenvalues $\\le\\lambda\\_0(\\tilde S)$, where $\\tilde S$ denotes the universal covering surface of *S*, endowed with the lifted Riemannian metric, and where $\\lambda\\_0(\\tilde S)$ denotes the bottom of the spectrum of $\\tilde S$. Recall here that, for a Riemannian surface *F* (possibly not complete) with piecewise smooth boundary ∂*F* (possibly empty), the *bottom of the spectrum of *F** is defined to be\n \n*λ*0 = *λ*0(*F*) = inf*R*(*φ*), \nwhere *φ* runs over all non-vanishing smooth functions on *F* with compact support in the interior $\\mathring F=F\\setminus\\partial F$ of *F* and where *R*(*φ*) denotes the Rayleigh quotient of *φ*. It is well known that the bottom of the spectrum of the Euclidean and hyperbolic plane is 0 and 1/4, respectively. When *F* is closed, *λ*0(*F*) = 0, when *F* is compact and connected with non-empty boundary, *λ*0(*F*) is the first Dirichlet eigenvalue of *F*. In the latter case, *λ*0(*F*)-eigenfunctions of *F* do not have zeros in $\\mathring F$ and, therefore, the multiplicity of *λ*0(*F*) as an eigenvalue of *F* is one. We then call the corresponding positive eigenfunction of *F* with *L*2-norm one the *ground state* of *F*.\nFor a Riemannian surface *S*, with or without boundary, we define the *analytic systole* to be the quantity\n \nΛ(*S*) = inf*F**λ*0(*F*), \nwhere the infimum is taken over all subsurfaces *F* in $\\mathring S$ with smooth boundary which are diffeomorphic to a closed disc, annulus, or cross cap. (A cross cap is frequently also called a Möbius strip.) Note that the fundamental groups of disc, annulus, and cross cap are cyclic, hence amenable. By the work of Brooks, we therefore have\n \nΛ(*S*) ≥ *λ*0(*S̃*) \nfor all complete and connected Riemannian surfaces *S*, see and also below. The strictness of this inequalilty and other estimates of Λ(*S*) are the topics of this article.\nTo clarify our terminology, a surface is a smooth manifold of dimension two. A Riemann surface is a surface together with a conformal structure. They are not the topic of this article. We study Riemannian surfaces, that is, surfaces together with a Riemannian metric.\nWe say that a surface *S* is of *finite type* if its Euler characteristic *χ*(*S*) is finite and its boundary is compact (possibly empty). It is well known that a connected surface *S* is of finite type if and only if *S* can be obtained from a closed surface by deleting a finite number of pairwise disjoint points and open discs.\nAfter first extensions of the results of Otal and Rosas in and, we showed in and that any complete Riemannian metric on a connected surface *S* of finite type with *χ*(*S*) < 0 has at most − *χ*(*S*) eigenvalues ≤ Λ(*S*), where the eigenvalues are understood to be Dirichlet eigenvalues if ∂*S* ≠ ∅. This result explains the significance of the analytic systole and the interest in establishing strictness in.\nStatement of main results\n-------------------------\nIn our first three results, we discuss the strictness of.\n[main1] If *S* is a compact and connected Riemannian surface whose fundamental group is not cyclic, then $\\Lambda(S)>\\lambda\\_0(\\tilde S)$.\nNote that the compact and connected surfaces with cyclic fundamental group are precisely sphere, projective plane, closed disc, closed annulus, and closed cross cap. For these, we always have equality in as we will see in.\nRecall that the spectrum of *S* is discrete if *S* is compact. In general, the spectrum of *S* is the disjoint union of its discrete and essential parts, where $\\lambda\\in\\R$ belongs to the essential spectrum of *S* if Δ − *λ* is not a Fredholm operator. The *bottom of the essential spectrum* is given by\n \n*λ*ess(*S*) = lim*K**λ*0(*S* \\ *K*), \nwhere *K* runs over the compact subsets of *S*, ordered by inclusion, see. By domain monotonicity, the limit is monotone with respect to the ordering of the compact subsets of *S*. If *S* is compact, then *λ*ess(*S*) = ∞. If *S* is a complete and connected Riemannian surface of finite type, then\n \n*λ*ess(*S*) ≥ Λ(*S*) \nsince the the ends of *S* admit neighborhoods in *S* whose connected components are diffeomorphic to open annuli. Such neighborhoods of the ends of *S* will be called *cylindrical*.\nextends in the following way to surfaces of finite type, be they compact or non-compact.\n[main3] If *S* is a complete and connected Riemannian surface of finite type whose fundamental group is not cyclic, then $\\Lambda(S)>\\lambda\\_0(\\tilde S)$ if and only if $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$.\nFor complete and connected Riemannian surfaces of finite type, Λ(*S*) is always between $\\lambda\\_0(\\tilde S)$ and *λ*ess(*S*), by and. Hence the condition $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$ in is obviously necessary to have the strict inequality $\\Lambda(S)>\\lambda\\_0(\\tilde S)$. The hard part of the proof of is to show that the condition is also sufficient.\nAnother way of stating the condition $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$ in is to require that there is a compact subset *K* of *S* such that *λ*0(*S* \\ *K*) > *λ*0(*S̃*). By domain monotonicity, this condition is then also satisfied for any compact subset *K*ʹ of *S* containing *K*.\n[exahype] Let *S* be a non-compact connected surface of finite type whose fundamental group is not cyclic. Using a decomposition of *S* into pairs of pants, it becomes obvious that *S* carries complete hyperbolic metrics with (possibly empty) geodesic boundary. For any such metric, we have\n \n$$\\lambda\\_{{\\operatorname{ess}}}(S) = \\Lambda(S) = \\lambda\\_0(\\tilde S) = 1/4$$ \nsince the ends of *S* are then hyperbolic cusps or funnels.\nTempted by this equality we investigate how *generic* this equality is among all smooth complete metrics on a non-compact surface of finite type. Our next result is that it is in fact rare. Some form of rigidity in the case of equality would of course be very interesting; compare with.\n[proqua] Let *S* be a connected surface of finite type. \n[1)]\n[proqua2] If the fundamental group of *S* is cyclic, then $\\Lambda(S)=\\lambda\\_0(\\tilde S)$ for any complete Riemannian metric on *S*. \n[proqua3] If *S* is non-compact and the fundamental group of *S* is not cyclic, then *S* carries complete Riemannian metrics such that *λ*ess(*S*) > Λ(*S*). Moreover, if *χ*(*S*) < 0, then such metrics may be chosen to have curvature *K* ≤ − 1 and finite or infinite area. \n[proqua4] If $\\lambda\\_0(\\tilde S, \\tilde g) > 0$ for some Riemannian metric *g* on *S*, then a generic complete Riemannian metric *g*ʹ on *S* in any neighborhood of *g* in the uniform *C*∞ topology satisfies the strict inequality *λ*ess(*S*, *g*ʹ) > Λ(*S*, *g*ʹ).\nBy, *λ*ess(*S*) > Λ(*S*) implies $\\Lambda(S)>\\lambda\\_0(\\tilde S)$.\nIn our next result, we generalize the main result of the third author in, which asserts that a hyperbolic metric on the closed surface *S**g* of genus *g* ≥ 2 has at most 2*g* − 2 eigenvalues ≤ 1/4 + *δ*, where\n \n*δ* = min{*π*/∣*S*∣, sys(*S*)2/∣*S*∣2}. \nHere ∣*S*∣ denotes the area of *S* and sys(*S*), the *systole* of *S*, is defined to be the minimal possible length of an essential closed curve in *S*.\n[main2] For a closed Riemannian surface *S* with curvature *K* ≤ *κ* ≤ 0, we have\n \n$$\\Lambda(S) \\ge -\\frac{\\kappa}4 + \\frac{{\\operatorname{sys}}(S)^2}{|S|^2}.$$ \n[1)]\nFor closed Riemannian surfaces *S* with curvature *K* ≤ *κ* < 0, we know in general only that *λ*0(*S̃*) ≥ − *κ*/4. Therefore may not imply the strict inequality Λ(*S*) > *λ*0(*S̃*) for such *S*. In fact, the relation between *λ*0(*S̃*) and the right hand side in is not clear. Our method of proof, involving isoperimetric inequalities and Cheeger’s inequality, does not seem to be sophisticated enough to capture the difference between them.\nThe proof of also applies to non-compact surfaces of finite type. In this case one needs to define the systole as the infimum over *all* homotopically non-trivial curves, not only the essential (not homotopic to a boundary component or a puncture) ones. For this reason, the corresponding statement is not really interesting anymore. If ∣*S*∣ < ∞, then sys(*S*) = 0 (by a refinement of the isosystolic inequality), and if ∣*S*∣ = ∞, then sys(*S*)/∣*S*∣ = 0.\nThe difference $\\Lambda(S)-\\lambda\\_0(\\tilde S)$ can not be estimated from below by a positive constant, which only depends on the topology and the area of *S*. In fact, given any $\\ve>0$ and natural number *n*, if the metric on *S* is hyperbolic with sufficiently small systole, then $\\lambda\\_n(S)<1/4+\\ve$, by or the proof of Theorem 8.1.2 in.\nOne may view also as an upper bound on the systole in terms of a curvature bound and Λ(*S*). Together with our next result, this explains the name *analytic systole*.\nFor a closed Riemannian surface *S*, we say that a closed geodesic *c* of *S* is a *systolic geodesic* if it is essential with length *L*(*c*) = sys(*S*). Clearly, systolic geodesics are simple.\n[esansy] If *S* is a closed Riemannian surface with *χ*(*S*) < 0 and curvature *K* ≥ − 1, then\n \n$$\\Lambda(S) \\le \\frac{1}{4} + \\frac{4\\pi^2}{w^2},$$ \nwhere\n \n$$w = w({\\operatorname{sys}}(S)) =\n \\begin{cases}\n \\operatorname{arsinh}(1/\\sinh({\\operatorname{sys}}(S)/2))) \\\\\n \\operatorname{arsinh}(1/\\sinh({\\operatorname{sys}}(S))) \\\\\n \\end{cases}$$ \nif *S* has a two-sided systolic geodesic or if all systolic geodesics of *S* are one-sided, respectively.\nHere we say that a simple closed curve in *S* is two-sided or one-sided if it has a tubular neigborhood which is diffeomorphic to an annulus or a cross cap, respectively.\nCombining and, we get that, for hyperbolic metrics, Λ(*S*) is squeezed between two functions of the systole.\n[syshype] For closed hyperbolic surfaces, we have\n \n$$\\frac{1}4 + \\frac{{\\operatorname{sys}}(S)^2}{4\\pi^2 \\chi(S)^2}\n \\le \\Lambda(S)\n \\le \\frac{1}4 + \\frac{4\\pi^2}{w^2}$$ \nwith *w* = *w*(sys(*S*)) as in.\nRecall that $\\operatorname{arsinh}x=\\ln(x+\\sqrt{x^2+1})$. In particular, we have\n \n$$w({\\operatorname{sys}}(S)) \\sim -\\ln({\\operatorname{sys}}(S)) \\to \\infty\n \\quad\\text{as ${\\operatorname{sys}}(S)\\to0$.}$$ \nWe conclude that the analytic systole of hyperbolic metrics on closed surfaces tends to 1/4 if and only if their systole tends to 0.\nMain problems and arguments\n---------------------------\nThe only surfaces *S* in and with Euler characteristic *χ*(*S*) ≥ 0 are torus and Klein bottle. For these, the proof of the inequality $\\Lambda(S)>\\lambda\\_0(\\tilde S)$ is quite elementary. The proof of the hard direction of, namely establishing the strict inequality $\\Lambda(S)>\\lambda\\_0(\\tilde S)$ under the condition $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$, is rather involved in the case *χ*(*S*) < 0.\nThe domain monotonicity of the first Dirichlet eigenvalue implies that Λ(*S*) can not be realized by any compact subsurface *F* ⊆ *S* diffeomorphic to a disc, an annulus or a cross cap. Keeping this in mind, our general strategy for the proof of is to show that the equality Λ(*S*) = *λ*0(*S̃*) would imply the existence of a non-trivial *λ*0(*S̃*)-eigenfunction $\\tilde\\vf$ on *S̃* or an appropriate cyclic quotient *Ŝ* of *S̃* that vanishes on an open set.\nThe condition $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$ forces a subsurface *F* with *λ*0(*F*) close to $\\lambda\\_0(\\tilde S)$ to stay almost completely in a large compact set in a weighted sense, the weight being the ground state. One then works essentially within a fixed compact subsurface of *S*. Two main problems that we still have to overcome in establishing the existence of $\\tilde\\vf$ as above are\n[1)]\na priori non-existence of a fixed quotient of $\\tilde S$ along a sequence of subsurfaces approximating *λ*0(*S̃*) and\nthe absence of the compact Sobolev embedding $H^1 \\hookrightarrow L^2$ on these covering spaces.\nAs for the first problem, the case of cross caps can be reduced to the case of annuli by considering the two-sheeted orientation covering of the original surface. The case of annuli is tackled by showing that only finitely many isotopy types of annuli have the bottom of their spectrum close to *λ*0(*S̃*). A keystone of the argument is which relates the bottom of the spectrum of compact surfaces *F* with the sum of the lengths of shortest curves in the free homotopy classes of the boundary circles of *F*.\nTo tackle the second problem, we establish, in, an inradius estimate for superlevel sets of suitably truncated ground states of a sequence of subsurfaces *F**n* approximating *λ*0(*S̃*). The inradius estimate is proved by means of isoperimetric inequalities, extending arguments from the proof of the Cheeger inequality.\nStructure of the article\n------------------------\nIn, we collect the relevant facts about isoperimetric inequalities on Riemannian surfaces. In, we extend Osserman’s refined version of the Cheeger inequality for plane domains to compact Riemannian surfaces with boundary. We also recall Osserman’s elegant proof since we will need consequences and extensions of his arguments. The isoperimetric inequalities from and the Cheeger inequality are then used in to obtain a generalized version of. The arguments here are very much in the spirit of Osserman and Croke. As an application of our discussion, we obtain for the case where *S* is a torus or a Klein bottle. This section closes with the proof of, which involves methods which are different from those of the rest of the article. Sections [secfe] and [secfe2] are concerned with properties of the ground states of compact Riemannian surfaces with boundary. The main objectives are on the relation of the bottom of the spectrum to other geometric quantities and on the inradius of superlevel sets of ground states. In, we complete the proof of. contains the proof of and some remarks and questions. In particular, we draw attention to problems in optimal design which are related to optimal estimates of the analytic systol. In, we discuss an extension of the result of Brooks quoted in connection with.\nIsoperimetric inequalities\n==========================\nThe content of the present section is related to and extends Lemma 1 of in the way we will need it.\nLet *F* be a compact and connected surface with piecewise smooth boundary ∂*F* ≠ ∅ and interior $\\mathring F=F\\setminus\\partial F$. The components of ∂*F* are piecewise smooth circles. Denote by *χ* = *χ*(*F*) the Euler characteristic of *F*.\nAssume that *F* is endowed with a Riemannian metric and denote by *K* the Gauss curvature of *F*. Let ∣*F*∣ and ∣∂*F*∣ be the area of *F* and the length of ∂*F*, respectively, and\n \n*ρ* = *ρ**F* = max{*d*(*x*, ∂*F*) ∣ *x* ∈ *F*} \nbe the *inradius* of *F*.\nFor a function $f \\colon F \\to \\R$, write *f*+ = max(*f*, 0) for its positive part. We recall the following isoperimetric inequalities.\n[l1] For any *F* as above and $\\kappa\\in\\R$, we have\n \n$$\\begin{aligned}\n |\\partial F|^2 &\\ge -\\kappa |F|^2 + 2\\left(2\\pi\\chi - \\int\\_F(K-\\kappa)^+dx \\right) |F|.\n \\label{l10}\\tag{1} \\intertext{If $\\kappa\\le0$, then}\n |\\partial F| &\\ge |F|{\\operatorname{ct}}\\_{\\kappa}\\rho\n + \\left(2\\pi\\chi - \\int\\_F(K-\\kappa)^+dx \\right){\\operatorname{tn}}\\_{\\kappa}\\frac\\rho2.\n \\label{l1a}\\tag{2}\\end{aligned}$$ \nIf *F* is not a disc and *κ* < 0, then\n \n$$(|\\partial F|^2-\\ell^2)^{1/2} \\ge \\sqrt{-\\kappa} |F|\n + \\frac1{\\sqrt{-\\kappa}}\\left(2\\pi\\chi - \\int\\_F(K-\\kappa)^+dx \\right).\n \\label{l1b}\\tag{3}$$ \nwhere ℓ denotes the sum of the lengths of the shortest loops in the free homotopy classes (in *F*) of the boundary circles of *F*.\nIn the second inequality, tn*κ* = sn*κ*/cs*κ* and ct*κ* = cs*κ*/sn*κ*, where sn*κ* and cs*κ* are the solutions of the differential equation $\\ddot u+\\kappa u=0$ with respective initial conditions\n \nsn*κ*(0) = 0, sn*κ*ʹ(0) = 1 and cs*κ*(0) = 1, cs*κ*ʹ(0) = 0. \nThe first inequality of corresponds to, the third to the (outer) inequality in (20) of. We added in *F* in parentheses in the statement since we will use in the case where *F* is a domain in a surface *S*. Then the length of a shortest loop in the free homotopy class in *S* of a boundary circle *c* of *F* might be smaller than the length of a shortest loop in the free homotopy class of *c* in *F*.\n[Proof of ([l1a])] We apply in the case *t* = *ρ*. The function *f* = *f*(*t*) of measures the area of the collar of width *t* about ∂*F* and, therefore, we have *f*(*ρ*) = ∣*F*∣ by the definition of *ρ*. The function *a* = *a*(*t*) of satisfies\n \n*a*(*ρ*) = *κ*∣*F*∣ − 2*π**χ* + ∫*F*(*K* − *κ*)+*d**x*. \nIn our notation, the function *ψ* of is given by\n \n$$\\psi(t) = a(t)\\frac{1-{\\operatorname{cs}}\\_\\kappa t}{\\kappa} + |\\partial F| {\\operatorname{sn}}\\_\\kappa t,$$ \nwhere we set (1 − cs*κ**t*)/*κ* = *t*2/2 for *κ* = 0. Now Theorem 2.4.2 of, in the case *κ* ≤ 0 and *t* = *ρ*, asserts that *f*(*ρ*) ≤ *ψ*(*ρ*), that is, that\n \n$$|F| \\le |F|(1-{\\operatorname{cs}}\\_\\kappa\\rho)\n - \\left( 2\\pi\\chi - \\int\\_F(K-\\kappa)^+dx \\right)\\frac{1-{\\operatorname{cs}}\\_\\kappa\\rho}{\\kappa}\n + |\\partial F|{\\operatorname{sn}}\\_{\\kappa}\\rho.$$ \nTherefore we get\n \n$$|\\partial F| {\\operatorname{sn}}\\_{\\kappa}\\rho\n \\ge |F|{\\operatorname{cs}}\\_{\\kappa}\\rho + \\left(2\\pi\\chi\n - \\int\\_F(K-\\kappa)^+dx \\right)\\frac{1-{\\operatorname{cs}}\\_{\\kappa}\\rho}{\\kappa}.$$ \nThis implies since (1 − cs*κ**t*)/*κ*sn*κ**t* = tn*κ*(*t*/2).\n[c1] If *K* ≤ *κ*, then we have:\n[1)]\n[c10] If *F* is a disc, then ∣∂*F*∣2 ≥ − *κ*∣*F*∣2 + 4*π*∣*F*∣.\n[c1a] If *χ* ≥ 0 and *κ* ≤ 0, then ∣∂*F*∣ ≥ ∣*F*∣ct*κ**ρ*.\n[c1b] If *χ* = 0 and *κ* ≤ 0, then ∣∂*F*∣2 ≥ − *κ*∣*F*∣2 + ℓ2.\nNote that we always have ∣∂*F*∣2 ≥ ℓ2, by the definition of ℓ.\nCheeger inequality revisited\n============================\nIn Lemma 2 of, Osserman discusses a refinement of the Cheeger inequality for compact planar domains, endowed with Riemannian metrics. We will need an extension of Osserman’s Lemma 2.\nAs above, we let *F* be a compact and connected Riemannian surface *F* with piecewise smooth boundary ∂*F* ≠ ∅. The *Cheeger constant* of *F* is defined to be the number\n \n*h* = *h*(*F*) = inf∣∂*F*ʹ∣/∣*F*ʹ∣, \nwhere the infimum is taken over all compact subsurfaces *F*ʹ of $\\mathring F$ with smooth boundary. Note that closed surfaces cannot occur as subsurfaces *F*ʹ of *F* since *F* is connected with non-empty boundary.\n[cheex] The Cheeger constant is given by\n \n*h* = inf∣∂*F*ʹ∣/∣*F*ʹ∣, \nwhere the infimum is taken over all compact and connected subsurfaces *F*ʹ of $\\mathring F$ with smooth boundary such that the boundary of each component of *F* \\ *F*ʹ has at least one boundary circle in $\\mathring F$ and contains at least one boundary circle of *F*.\nLet *F*ʹ be a compact subsurface of $\\mathring F$ with smooth boundary and denote by *F*ʹ1, …, *F*ʹ*k* the components of *F*ʹ. Then\n \n∣*F*ʹ∣ = ∣*F*ʹ1∣ + … + ∣*F*ʹ*k*∣ and ∣∂*F*ʹ∣ = ∣∂*F*ʹ1∣ + … + ∣∂*F*ʹ*k*∣ \nand hence\n \n$$\\inf\\frac{|\\partial F'\\_i|}{|F'\\_i|} \\le \\frac{|\\partial F'\\_1|+\\dots+|\\partial F'\\_k|}{|F'\\_1|+\\dots+|F'\\_k|} = \\frac{|\\partial F'|}{|F'|}.$$ \nThis shows that the infimum *h* can be taken over compact and connected subsurfaces *F*ʹ of $\\mathring F$ with smooth boundary.\nLet now *C* be a component of *F* \\ *F*ʹ. Suppose first that the boundary of *C* does not contain a boundary circle of *F*. Then *F*ʺ = *F*ʹ ∪ *C* is a compact and connected subsurface of $\\mathring F$ with area ∣*F*ʺ∣ > ∣*F*ʹ∣ and length of boundary ∣∂*F*ʺ∣ < ∣∂*F*ʹ∣. It follows that the infimum *h* is attained by compact and connected subsurfaces *F*ʹ of $\\mathring F$ with smooth boundary such that the boundary of each component of *F* \\ *F*ʹ contains a boundary circle of *F*.\nIf the boundary of *C* would not have a boundary circle in $\\mathring F$, then *C* would have to coincide with *F* since *F* is connected. But then *F*ʹ would be empty, a contradiction.\nBy a slight variation of the standard terminology, we say that a subsurface *S* of a surface *T* is *incompressible in *T** if the induced maps of fundamental groups are injective, for all connected component *C* of *S*. In particular, embedded discs are always incompressible.\n[cheey] The Cheeger constant is given by\n \n*h* = inf∣∂*F*ʹ∣/∣*F*ʹ∣, \nwhere the infimum is taken over all incompressible compact and connected subsurfaces *F*ʹ of $\\mathring F$ with smooth boundary such that no component of *F* \\ *F*ʹ is a disc or a cross cap. Any such *F*ʹ satisfies *χ*(*F*ʹ) ≥ *χ*(*F*) with equality if and only if *F* \\ *F*ʹ is a collared neighborhood of ∂*F*, consisting of annuli about the boundary circles of *F*.\nFor example, if *F* is an annulus, then we only need to consider discs and incompressible annuli *F*ʹ in *F*; if *F* is a cross cap, then only discs and incompressible annuli and cross caps *F*ʹ.\n[Proof of ] By, the Cheeger constant *h* is realized by compact and connected subsurfaces *F*ʹ of $\\mathring F$ with smooth boundary such that each component of *F* \\ *F*ʹ has at least two boundary circles. This excludes discs and cross caps as components of *F* \\ *F*ʹ. We have\n \n$$\\chi(F) = \\chi(F') + \\chi(F\\setminus\\mathring F') \\le \\chi(F')$$ \nsince the intersection of *F*ʹ with $F\\setminus\\mathring F'$ consists of circles and since no component of $F\\setminus\\mathring F'$ is a disc. Furthermore, equality can only occur if $\\chi(F\\setminus\\mathring F')=0$. By what we already know, this can only happen if the components of $F\\setminus\\mathring F'$ are annuli. By and since $F'\\subseteq\\mathring F$, they constitute a collared neighborhood of ∂*F*.\nIt remains to show the incompressibility of *F*ʹ. If this would not hold, *F*ʹ would contain a Jordan loop *c* which is not contractible in *F*ʹ, but is contractible in $\\mathring F$. Then *c* would be the boundary of an embedded disc *D* in $\\mathring F$ which is not contained in *F*ʹ. Since ∂*D* ⊆ *F*ʹ, *D* \\ *F*ʹ would consist of components of *F* \\ *F*ʹ. Their boundary would be in $D\\subseteq\\mathring F$ in contradiction to.\nRecall the classical Cheeger inequality.\n[Cheeger inequality][cheei] We have *λ*0(*F*) ≥ *h*2/4.\nIn the proofs of and, we will need arguments and consequences of the proof of and, therefore, recall the elegant arguments from the proof of the corresponding Lemma 2 in.\n[Recalling the proof of the Cheeger inequality] Since *F* is compact with piecewise smooth boundary, *λ*0 = *λ*0(*F*) is the first Dirichlet eigenvalue of *F*. Let *φ* be the corresponding ground state and set *ψ* = *φ*2. By the Schwarz inequality, we have\n \n$$\\label{chee1}\n\\begin{split}\n \\int\\_F |\\nabla \\psi|\n &= \\int\\_F 2 |\\varphi| |\\nabla \\varphi|\n \\leq 2 \\left(\\int\\_F|\\nabla \\varphi|^2 \\right)^{1/2} \\left(\\int\\_F \\varphi^2 \\right)^{1/2}\n \\\\\n &= 2 \\sqrt{\\lambda\\_0} \\int\\_F \\varphi^2\n = 2 \\sqrt{\\lambda\\_0} \\int\\_F \\psi.\n\\end{split}$$ \nThis implies\n \n$$\\label{chee2}\n \\sqrt{\\lambda\\_0} \\ge \\frac12\\frac{\\int\\_F|\\nabla\\psi|}{\\int\\_F\\psi}.$$ \nFor regular values *t* > 0 of *ψ*, let *F**t* = {*ψ* ≥ *t*} and denote by *A*(*t*) and *L*(*t*) the area and the length of *F**t* and ∂*F**t* = {*ψ* = *t*}, respectively. For the null set of singular values of *ψ*, set *A*(*t*) = *L*(*t*) = 0. The coarea formula gives\n \n∫*F*∣∇*ψ*∣ = ∫0∞*L*(*t*)*d**t*. \nOn the other hand, since ∫*F**ψ* computes the volume of the domain\n \n$$\\{(x,y)\\in F\\times\\R \\mid 0\\le y\\le \\psi(x) \\},$$ \nCavalieri’s principle gives\n \n∫*F**ψ* = ∫0∞*A*(*t*)*d**t*. \nBy the definition of *h* = *h*(*F*), we have\n \n∫*F*∣∇*ψ*∣ = ∫0∞*L*(*t*)*d**t* ≥ *h*∫0∞*A*(*t*)*d**t* = *h*∫*F**ψ*. \nCombining and, we get *λ*0 ≥ *h*2/4 as asserted.\nQuantitative estimates of Λ(*S*)\n================================\nWe start with a version of for surfaces with (possibly empty) boundary.\n[mainb] Let *S* be a compact and connected Riemannian surface, with or without boundary, with infinite fundamental group and curvature *K* ≤ *κ*, where *κ* is a constant. Then we have: \n[1)]\n[mainb1] If *κ* ≤ 0, then\n \n$$\\Lambda(S) \\ge -\\frac{\\kappa}4 + \\frac1{|S|}\\min\\left\\{\\pi,\\frac{{\\operatorname{sys}}(S)^2}{|S|}\\right\\}.$$ \n[mainb2] If *κ* > 0 and *S* is orientable, then\n \n$$\\Lambda(S) \\ge \\min\\left\\{\\frac{\\pi}{|S|}-\\frac{\\kappa}4,\\frac{{\\operatorname{sys}}(S)^2}{|S|^2}\\right\\}.$$ \n[mainb3] If *κ* > 0 and *S* is non-orientable, then\n \n$$\\Lambda(S) \\ge \\min\\left\\{\\frac{\\pi}{|S|}-\\frac{\\kappa}4,\\frac{{\\operatorname{sys}}(S)^2}{4|S|^2}\\right\\}.$$ \nFor a closed disc *D* in *S*,.[c10] implies that\n \n$$\\label{ched}\n \\frac{|\\partial D|^2}{|D|^2} \\ge -\\kappa + \\frac{4\\pi}{|D|} \\ge -\\kappa + \\frac{4}{|S|}\\pi.$$ \nSuppose now that *A* is a closed annulus in *S*. Suppose first that the boundary circles of *A* are null-homotopic in *S*. Then by the Schoenflies theorem (see also ) there is a disc *D* in $S\\setminus\\mathring A$ such that *F*ʹ = *A* ∪ *D* is a disc. Then ∣∂*F*ʹ∣ ≤ ∣∂*A*∣ and ∣*F*ʹ∣ ≥ ∣*A*∣. Using.[c10] again, we get that also holds for *A* in place of *D*.\nAssume now that the boundary circles of *A* are not null-homotopic in *S*. By.[c1b] and the statement after it, we have\n \n∣∂*A*∣2 ≥ − min(*κ*, 0)∣*A*∣2 + 4*l*(*A*)2, \nwhere *l*(*A*) denotes the length of a shortest curve in the free homotopy class in *A* of the two boundary circles of *A*. Since the boundary circles of *A* are not homotopic to zero in *S*, we have *l*(*A*) ≥ sys(*S*). Hence\n \n$$\\label{chea}\n \\frac{|\\partial A|^2}{|A|^2} \\ge -\\min(\\kappa,0) + 4\\frac{{\\operatorname{sys}}(S)^2}{|A|^2}\n \\ge -\\min(\\kappa,0) + \\frac{4}{|S|}\\frac{{\\operatorname{sys}}(S)^2}{|S|}.$$ \nIf *C* is a cross cap in *S*, then *S* is not orientable. Now the soul of *C* is not homotopic to zero in *S* and the fundamental group of *S* is torsion free. Since the boundary circle ∂*C* of *C* is freely homotopic to the soul of *C*, run twice, we get that ∂*C* is not homotopic to zero in *S*. In particular, we always have ∣∂*C*∣ ≥ sys(*S*). If *κ* ≤ 0, then a shortest curve in *S* in the free homotopy class of the soul of *C*, run twice, is a shortest curve in *S* in the free homotopy class of the boundary circle of *C*. Hence ∣∂*C*∣ ≥ 2sys(*S*) if *κ* ≤ 0. We conclude that also holds for *C* in place of *A* if *κ* ≤ 0. In the general case, ∣∂*C*∣ ≥ sys(*S*) implies a modified version of with *C* in place of *A*, where the factor 4 on the right hand side is replaced by 1.\nNow the assertions of follows from the Cheeger inequality () in combination with,, and or the modified version of, respectively.\n[Proof of ] It remains to show that sys(*S*)2/∣*S*∣ ≤ *π* if *S* is closed with curvature *K* ≤ 0. In fact, in that case, the injectivity radius of *S* is sys(*S*)/2. Then the exponential map exp*p* at any point *p* ∈ *S* is a diffeomorphism from the disc of radius sys(*S*)/2 in *T**p**S* to its image, the metric ball *B* = *B*(*p*, sys(*S*)/2) about *p* in *S*. By comparison with the flat case, we get ∣*B*∣ ≥ *π*sys(*S*)2/4 and therefore\n \n$${\\operatorname{sys}}(S)^2/|S| < {\\operatorname{sys}}(S)^2/|B| \\le 4/\\pi.\n \\qedhere$$ \n[mainc]\n[1)]\nIf *S* is a compact and connected surface with non-empty boundary, then *S* contains a finite graph *G* in its interior which is a deformation retract of *S*. Given a Riemannian metric on S, a sufficiently small tubular neighborhood *T* of *G* in *S* is a Riemannian surface diffeomorphic to *S* with sys(*T*) ≥ sys(*S*) and with arbitrarily small area. Moreover, any upper bound on the curvature persists. In other words, we cannot expect to remove the minimum on the right hand side of the estimates in.\nNote also that the right hand side of the inequalities in [mainb2]) and [mainb3]) of is positive if and only if ∣*S*∣ < 4*π*/*κ*, that is, if and only if ∣*S*∣ is smaller than the area of the sphere of constant curvature *κ* > 0.\nIn, Gromov shows that sys(*S*)2/∣*S*∣2 ≤ 4/3 for any closed Riemannian surface. The point is, of course, that his estimate is curvature free. His work in also implies that \nsys(*S**g*)2/∣*S**g*∣2 ≤ *C**g*(ln*g*)2/*g* \n with limsup*C**g* ≤ 1/*π* as *g* → ∞; see Section 11.3 in.\n[Proof of in the case *χ*(*S*) ≥ 0] In view of, it remains to show that $\\Lambda(S)>\\lambda\\_0(\\tilde S)$ in the case where *S* is a torus or a Klein bottle. Then *S* admits a flat background metric *h* which is conformal to the given metric *g* of *S*. applies to *h* and shows that\n \nΛ(*S*, *h*) ≥ sys(*S*, *h*)2/∣(*S*, *h*)∣2, \nwhere (*S*, *h*) denotes *S*, endowed with the metric *h*. Furthermore, since we are in the case of surfaces, the Dirichlet integral of smooth functions is invariant under conformal changes; that is, we have\n \n$$\\label{dirich}\n \\int |\\nabla\\vf|^2\\,da = \\int |\\nabla\\vf|\\_h^2\\,da\\_h.$$ \nSince *S* is compact, there is a constant *α* ≥ 1 such that\n \n*α*− 1∣*v*∣ ≤ ∣*v*∣*h* ≤ *α*∣*v*∣ \nfor all tangent vectors *v* of *S*. Using and, we obtain\n \nΛ(*S*) ≥ *α*− 2Λ(*S*, *h*) ≥ *α*− 8sys(*S*)2/∣*S*∣2 > 0. \nOn the other hand, the fundamental group of *S* is amenable and hence $\\lambda\\_0(\\tilde S)=\\lambda\\_0(S)$ by. Now *S* is a torus or a Klein bottle, hence $\\lambda\\_0(\\tilde S)=0$.\n[Proof of ] Suppose first that *S* is orientable, that is, that *S* = *S**g* for some *g* ≥ 2, and let *c* be a systolic geodesic on *S*. Then by, the tubular neighborhood *T* of *c* of width\n \n*w*2 = arsinh(1/sinh(sys(*S*)/2))) \nis an open annulus. Since *c* is essential, *T* is incompressible. Note also that *T* can be exhausted by incompressible compact annuli with smooth boundary. In particular, for any *r* < *w*2, the closed metric ball $\\bar B(p,r)$ of radius *r* about a point *p* on *c* is contained in an incompressible compact annulus *A**r* ⊆ *T* with smooth boundary. Since *B*(*p*, *r*) ⊆ *A**r*, we may use Theorem 1.1 and the first displayed formula on page 294 of to conclude that\n \n$$\\lambda\\_0(A\\_r) \\le \\lambda\\_0(\\bar B(p,r)) \\le - \\frac{\\kappa}{4} + \\frac{4\\pi^2}{r^2}.$$ \nBy the definition of Λ(*S*), we have Λ(*S*) ≤ *λ*0(*A**r*) for any *r* as above. Hence the claim of follows in the case *S* = *S**g*.\nSuppose now that *S* is not orientable. Let Or(*S*) → *S* be the orientation covering of *S* and *c* be a systolic geodesic on *S*. There are two cases:\n[1)]\n[one] If *c* is one-sided, then the lift $\\tilde c$ of *c* to Or(*S*) is simple of length 2*L* and is invariant under the non-trivial covering transformation *f* of Or(*S*). Again by, the tubular neighborhood *T* of $\\tilde c$ of width\n \n*w*1 = arsinh(1/sinh(sys(*S*))) \nis an open annulus. Since *f* leaves $\\tilde c$ invariant, it also leaves *T* invariant and *T*/*f* is an open cross cap with soul *c* and width *w*1 about *c*. Hence for any *r* < *w*1, the closed metric ball $\\bar B(p,r)$ of radius *r* about a point *p* on *c* is contained in a compact cross cap *C**r* ⊆ *T*/*f* with smooth boundary.\n[two] If *c* is two-sided, then *c* has two lifts *c*1 and *c*2 to Or(*S*) and both are simple of length *L*. Moreover, by, the tubular neighborhoods *T*1 of *c*1 and *T*2 of *c*2 of width *w*2 are open annuli and do not intersect. Now *f* permutes *c*1 and *c*2, therefore also *T*1 and *T*2, and hence the tubular neighborhood *T* of *c* of width *w*2 is an open annulus.\nIn both cases, [one]) and [two]), we can now conclude the proof of the claim of as in the case *S* = *S**g*.\n[eslam] The arguments in the proof of also show that diam*S* ≥ *w* with *w* = *w*2 and *w* = *w*1, respectively. Hence we get\n \n$$\\lambda\\_{-\\chi(S)} \\le -\\frac{\\kappa}{4} + \\chi(S)^2\\frac{16\\pi^2}{w^2}$$ \nfrom Corollary 2.3 of. In view of Λ(*S*) < *λ*− *χ*(*S*), this gives another, but weaker upper bound for Λ(*S*).\nOn the ground state\n===================\nThroughout this section, we let *F* be a compact Riemannian surface with smooth boundary ∂*F* ≠ ∅ and $\\vf$ be the ground state of *F*. We also set $\\psi=\\vf^2$ and let *F**t* = {*ψ* ≥ *t*}. Note that ∫*ψ* = 1.\nBy the Hopf boundary lemma, $\\vf$ does not have critical points on ∂*F*. Moreover, since $\\vf>0$ in the interior $\\mathring F$ of *F*, a point in $\\mathring F$ is critical for *ψ* if and only if it is critical for $\\vf$. All points of ∂*F* are critical for *ψ*.\nIn our first result, we elaborate on the argument from the middle of page 549 in.\n[topft] Let 0 < *t* < max*ψ* be a regular value of *ψ*. Then *F**t* is a compact subsurface of $\\mathring F$ such that the boundary of each component of *F* \\ *F**t* has at least one boundary circle in $\\mathring F$ and contains at least one boundary circle of *F*.\nSince *t* is a regular value of *ψ* with 0 < *t* < max*ψ*, *F**t* is a compact subsurface of $\\mathring F$ with smooth boundary. If the boundary of a component *C* of *F* \\ *F**t* would not contain a boundary circle of *F*, then *φ* would be a non-constant superharmonic function on *C* which attains its maximum $\\sqrt t$ along ∂*F**t*, a contradiction. Clearly, the boundary of *C* must have at least one boundary circle in $\\partial F\\_t\\subseteq\\mathring F$.\n[topft2] Let 0 < *t* < max*ψ* be a regular value of *ψ*. Then any connected component *C* of *F**t* is incompressible in *F* and no component of *F* \\ *C* is a disc or a cross cap. Furthermore, *χ*(*C*) ≥ *χ*(*F*) with equality if and only if *F* \\ *C* is a collared neighborhood of ∂*F*, consisting of annuli about the boundary circles of *F*.\nSuppose that a component *D* of *F* \\ *C* would be a disc or a cross cap. Then *D* \\ *F**t* would consist of components of *F* \\ *F**t* with boundary in $\\mathring F$, a contradiction to. Substituting *C* for *F*ʹ, the rest of the proof of is now more or less the same as that of.\nAs in, we denote by ℓ the sum of the lengths of the shortest loops in the free homotopy classes (in *F*) of the boundary circles of *F*. Furthermore, we let Λʹ(*F*) = inf*λ*0(*F*ʹ), where the infimum is taken over all incompressible compact and connected subsurfaces *F*ʹ of $\\mathring F$ with smooth boundary and Euler characteristic *χ*(*F*ʹ) > *χ*(*F*).\n[lambdad] If *χ*(*F*) ≤ 0, then\n \n$$\\lambda\\_0(F)\\ge \\big\\{1-\\delta\n + 2\\big(1-\\frac1\\delta\\big)\\frac{|F|}{\\ell}\\sqrt{\\lambda\\_0(F)}\\big\\}\\Lambda'(F)$$ \nfor all 0 < *δ* < 1/2.\nSince the quantities involved in the lemma vary continuously with respect to variations of the metric (in the *C*0-topology), we may assume, by Theorem 8 in, that *φ* is a Morse function. Then the critical points of *ψ* in $\\mathring F$ are non-degenerate. Moreover, since $\\vf$ does not have critical points on ∂*F*, *F* \\ *F**t* is a collared neighborhood of ∂*F*, consisting of annuli about the boundary circles of *F*, for all sufficiently small *t* > 0. On the other hand, for *t* < max*ψ* suffciently close to max*ψ*, *F**t* is a union of embedded discs, one for each maximum point of *ψ*. Hence the topology of *F**t* undergoes changes as *t* increases from 0 to max*ψ*.\nSince $\\vf$ is a Morse function, *ψ* has only finitely many critical points in $\\mathring F$. By, *ψ* does not have local minima in $\\mathring F$. Hence critical points of *ψ* in $\\mathring F$ are saddle points and local maxima.\nLet 0 = *β*0 < … < *β**m* = max*ψ* be the finite sequence of critical values of *ψ* and choose $\\ve>0$ with $\\ve<\\min\\{\\beta\\_{i+1}-\\beta\\_i\\}$. In a first step, we select now a critical value *β* = *β**i* according to specific requirements.\nBy, each component *C* of $F\\_{\\beta\\_1+\\ve}$ has Euler characteristic *χ*(*C*) ≥ *χ*(*F*). Therefore there are two cases. Either each component *C* of $F\\_{\\beta\\_1+\\ve}$ has Euler characteristic *χ*(*C*) > *χ*(*F*). Then we set *β* = *β*1. Or else there is a component *C* with *χ*(*C*) = *χ*(*F*). Then *F* \\ *C* is a collared neighborhood of ∂*F*, consisting of annuli about the boundary circles of *F*, by. In that case, by, the other components of $F\\_{\\beta\\_1+\\ve}$ are discs contained in these annuli.\nWe assume that we are in the second case and consider the second critical value *β*2. By, there are again two cases. Either each component *C* of $F\\_{\\beta\\_2+\\ve}$ has Euler characteristic *χ*(*C*) > *χ*(*F*); then we set *β* = *β*2. Or else there is a component *C* of $F\\_{\\beta\\_2+\\ve}$ with *χ*(*C*) = *χ*(*F*). Then *F* \\ *C* is a collared neighborhood of ∂*F* consisting of annuli about the boundary circles of *F*. In the latter case, we pass on to the next critical value *β*3. Since *χ*(*F*) ≤ 0, we will eventually arrive at a first critical value *β* = *β**i* with the property that the complement of a component of $F\\_{\\beta-\\ve}$ is a collared neighborhood of ∂*F* consisting of annuli about the boundary circles of *F* and such that each component *C* of $F\\_{\\beta+\\ve}$ has Euler characteristic *χ*(*C*) > *χ*(*F*). Note that this property then holds for all sufficiently small $\\ve>0$ since *β* is the only critical value of $\\vf$ in (*β**i* − 1, *β**i* + 1). It follows that for any regular value 0 < *t* < *β* of *ψ*, *F**t* has a component *C* such that *F* \\ *C* is a collared neighborhood of ∂*F* consisting of annuli about the boundary circles of *F*. In particular, ∣∂*F**t*∣ ≥ ℓ for all such *t*. Using and, we obtain\n \n$$\\label{cheeta5}\n \\beta \\ell \\le \\int\\_0^\\infty L(t) dt \\le 2\\sqrt{\\lambda\\_0(F)},$$ \nwhere *L*(*t*) denotes the length of ∂*F**t*.\nFor $\\ve > 0$ as above, the smooth function $\\vf\\_\\ve=\\varphi -\\sqrt{\\beta + \\ve}$ is smooth on $F\\_{\\beta+\\ve},$ vanishes on $\\partial F\\_{\\beta+\\ve}$ and satisfies\n \n$$\\label{terms}\n\\begin{split}\n \\int\\_{F\\_{\\beta+\\ve}}\\vf\\_\\ve^2\n &= \\int\\_{F\\_{\\beta+\\ve}}\\big(\\varphi - \\sqrt{\\beta + \\ve}\\big)^2 \\\\\n &= \\int\\_{F\\_{\\beta+\\ve}}\\varphi^2 - 2 \\sqrt{\\beta + \\ve}\n \\int\\_{F\\_{\\beta+\\ve}}\\varphi + (\\beta + \\ve) |F\\_{\\beta+\\ve}|.\n\\end{split}$$ \nNow the first term on the right hand side of satisfies\n \n$$\\label{terms2}\n \\int\\_{F\\_{\\beta+\\ve}}\\varphi^2 \\ge \\int\\_F\\vf^2 -(\\beta+\\ve)(|F|-|F\\_{\\beta+\\ve}|)$$ \nsince $\\vf^2\\le\\beta+\\ve$ on $F\\setminus F\\_{\\beta+\\ve}$. For the second term on the right hand side of, we have\n \n$$\\label{terms4}\n\\begin{split}\n 2 \\sqrt{\\beta + \\ve} \\int\\_{F\\_{\\beta+\\ve}}\\varphi\n &\\le 2\\sqrt{\\beta+\\ve}|F\\_{\\beta+\\ve}|^{1/2}\\big(\\int\\_{F\\_{\\beta+\\ve}}\\vf^2\\big)^{1/2} \\\\\n &\\le \\frac1\\delta(\\beta+\\ve)|F\\_{\\beta+\\ve}| + \\delta \\int\\_{F\\_{\\beta+\\ve}}\\varphi^2 \\\\\n &\\le \\frac1\\delta(\\beta+\\ve)|F\\_{\\beta+\\ve}| + \\delta \\int\\_{F}\\varphi^2\n\\end{split}$$ \nby the Schwarz inequality and the Peter and Paul principle. Combining,, and and using that the *L*2-norm of $\\vf$ is one and that 2 − 1/*δ* < 0, we obtain\n \n$$\\begin{aligned}\n \\int\\_{F\\_{\\beta+\\ve}}\\vf\\_\\ve^2 \\notag\n &\\ge (1-\\delta) \\int\\_F\\varphi^2 -(\\beta+\\ve)|F| + \\big(2-\\frac1\\delta\\big)(\\beta+\\ve)|F\\_{\\beta+\\ve}| \\\\\n &\\ge 1-\\delta + \\big(1-\\frac1\\delta\\big)(\\beta+\\ve)|F|.\\end{aligned}$$ \nFor the Rayleigh quotient of $\\vf\\_\\ve$, we get\n \n$$\\begin{aligned}\n &R(\\vf\\_\\ve)(1-\\delta + \\big(1-\\frac1\\delta\\big)(\\beta+\\ve)|F|)\n \\le R(\\vf\\_\\ve)\\int\\_{F\\_{\\beta+\\ve}}\\vf\\_\\ve^2 \\\\\n &\\hspace{19mm}\n = \\int\\_{F\\_{\\beta+\\ve}} |\\nabla \\vf\\_\\ve|^2\n = \\int\\_{F\\_{\\beta+\\ve}} |\\nabla \\varphi|^2\n \\le \\int\\_F|\\nabla \\varphi|^2\n = \\lambda\\_0(F).\\end{aligned}$$ \nSince $F\\_{\\beta + \\ve}$ is a disjoint union of incompressible compact and connected subsurfaces *F*ʹ with smooth boundary and *χ*(*F*ʹ) > *χ*(*F*), we also have $R(\\vf\\_\\ve) \\ge \\inf\\Lambda'(F)$. Letting $\\ve$ tend to 0, we finally obtain\n \n$$\\label{lambdabe}\n \\Lambda'(F)(1-\\delta + \\big(1-\\frac1\\delta\\big)\\beta|F|)\n \\le \\lambda\\_0(F).$$ \nCombining and, we arrive at.\nOn the ground state (continued)\n===============================\nLet *S* be a complete and connected Riemannian surface of finite type with *χ*(*S*) < 0 and $\\lambda\\_0(\\tilde S)<\\lambda\\_{\\rm ess}(S)$.\nSince *χ*(*S*) < 0, *S* carries complete hyperbolic metrics. Using a decomposition of *S* into a finite number of pairs of pants, it is clear that we may choose such a metric *h* such that the connected components of a neighborhood of the ends of *S* is a finite union of *hyperbolic funnels*, that is, cylinders of the form $(-1,\\infty)\\times\\R/\\Z$ with metric\n \n*d**r*2 + cosh(*r*)2*d**ϑ*2. \nThen the curves *r* = 0 are closed *h*-geodesics of length 1. The original metric of *S* will be denoted by *g*.\nWe fix a smooth and proper function from *S* to [0, ∞), which agrees outside a compact set with the coordinates *r* in each of the ends. By abuse of notation, we denote this function by *r*. Choose an increasing sequence 0 < *r*0 < *r*1 < *r*2 < … → ∞. Then the subsurfaces\n \n*K**i* = {*r* ≤ *r**i*} \nof *S* are compact with smooth boundary ∂*K**i* = {*r* = *r**i*} such that *S* \\ *K**i* is a cylindrical neighbourhood of infinity. Furthermore,\n \n*K*0 ⊆ *K*1 ⊆ *K*2 ⊆ … \nis an exhaustion of *S*. By choosing the sequence of *r**i* suitably, we may assume that \n[a)]\nthere exist cutoff functions *η**i*: *S* → [0, 1] with *η**i* = 1 on *K**i*, *η**i* = 0 outside *K**i* + 1, and ∣∇*η**i*∣2 ≤ 1/*i*, \n$\\lambda\\_0(\\tilde S) < \\lambda\\_0(S\\setminus K\\_0)$, \nwhere we note that $\\lambda\\_0(S\\setminus K\\_i)<\\lambda\\_0(S\\setminus K\\_{i+1})\\cdots\\to\\lambda\\_{\\rm ess}(S)$.\nIn the case where *S* is compact, we have *K**i* = *S* for all *i* and part of the following discussion becomes trivial.\nWe now let *F* be a compact subsurface of *S* with smooth boundary ∂*F* ≠ ∅. As in, we denote by $\\vf$ the ground state of *F* and let $F\\_t=\\{\\vf^2\\ge t\\}$.\n[vfreg] For a subset $\\mathcal R\\subseteq(0,\\max\\vf^2)$ of full measure, *F**t* is a smooth subsurface of $\\mathring F$ such that $\\partial F\\_t=\\{\\vf=t\\}$ and ∂*K**i* intersect transversally for all *i*.\nSince $\\vf$ is smooth up to the boundary of *F* and has no critical points on ∂*F*, there is a smooth extension $\\tilde\\vf$ of $\\vf$ to *S* such that $\\tilde\\vf$ is strictly negative on *S* \\ *F*. The restriction $\\tilde\\vf\\_0$ of $\\tilde\\vf$ to the union of the curves {*r* = *r**i*} is then smooth, and hence there is a set $\\mathcal R\\_0\\subseteq\\R$ of full measure such that any $t\\in\\mathcal R\\_0$ is a regular value of $\\tilde\\vf\\_0$. Note that $\\nabla\\tilde\\vf$ is not perpendicular to the curve {*r* = *r**i*} at points *p* ∈ {*r* = *r**i*} with $\\tilde\\vf(p)\\in\\mathcal R\\_0$. On the other hand, $\\nabla\\tilde\\vf$ is perpendicular to ∂*F**t* for any regular value *t* of $\\vf^2$ in $(0,\\max\\vf^2)$. Therefore the intersection $\\mathcal R$ of $\\mathcal R\\_0$ with the set of regular values of $\\vf^2$ in $(0,\\max\\vf^2)$ satisfies the required assertions.\n[topft3] For any $t\\in\\mathcal R$, the intersection *F**t* ∩ *K**i* is a subsurface of *F* with piecewise smooth boundary and any connected component *C* of *F**t* ∩ *K**i* is incompressible in *F*. In particular, we have *χ*(*C*) ≥ *χ*(*F*).\nFor any $t\\in\\mathcal R$, $\\partial F\\_t=\\{\\vf=t\\}$ and ∂*K**i* intersect transversally for all *i*, and then *F**t* ∩ *K**i* is a subsurface of *F* with piecewise smooth boundary. Since *S* \\ *K**i* is a cylindrical neighborhood of the ends of *S*, a disc in *S* has to be contained in *K**i* if its boundary is in *K**i*. Hence the components of *F**t* ∩ *K**i* are incompressible in *F**t*. By, *F**t* is incompressible in *F*. Therefore *F**t* ∩ *K**i* is incompressible in *F*.\n[comploc] Assume that *λ*0(*F*) ≤ *θ**λ*0(*S* \\ *K*0) for some 0 < *θ* < 1, and let $\\ve>0$. Then there is an integer $i\\_0=i\\_0(\\theta,\\ve)\\ge0$ such that\n \n$$\\int\\_{F\\cap K\\_i} \\vf^2 \\ge 1-\\ve\n \\quad\\text{for all $i\\ge i\\_0$.}$$ \nSince $(1-\\eta\\_i) \\vf$ has support in *F* \\ *K**i* ⊆ *S* \\ *K*0, we have\n \n$$\\begin{aligned}\n \\lambda\\_0(S\\setminus K\\_0) \\int\\_F (1-\\eta\\_i)^2 & \\vf^2\n \\leq \\int\\_F |\\nabla ( (1-\\eta\\_i)\\vf)|^2\n \\\\\n &= \\int\\_F \\nabla ((1-\\eta\\_i)^2\\vf) \\cdot \\nabla \\vf + \\int\\_F \\vf^2 |\\nabla(1-\\eta\\_i)|^2\n \\\\ \n &= \\int\\_F ((1-\\eta\\_i)^2\\vf) \\cdot \\Delta\\vf + \\int\\_F \\vf^2 |\\nabla(1-\\eta\\_i)|^2\n \\\\ \n &\\leq \\lambda\\_0(F) \\int\\_F (1-\\eta\\_i)^2 \\vf^2 + \\frac{1}{i} \\int\\_F \\vf^2\n \\\\\n &\\leq \\theta \\lambda\\_0(S\\setminus K\\_0) \\int\\_F(1-\\eta\\_i)^2 \\vf^2 + \\frac{1}{i}.\\end{aligned}$$ \nSince 0 < *θ* < 1, we conclude that\n \n$$\\int\\_{F} (1-\\eta\\_i)^2 \\vf^2 \\leq \\frac{1}{(1-\\theta)\\lambda\\_0(S\\setminus K\\_0)i}.$$ \nNow for *i*0 sufficiently large, the right hand side is smaller than $\\ve$ for all *i* ≥ *i*0 − 1. For any *i* ≥ *i*0, we then have\n \n$$\\begin{split}\n \\int\\_{F\\cap K\\_{i}} \\vf^2 &= 1-\\int\\_{F\\setminus K\\_{i}} \\vf^2\n \\\\\n &= 1- \\int\\_{F\\setminus K\\_{i}} (1-\\eta\\_{i-1})^2 \\vf^2 \n \\\\\n &\\geq 1 - \\int\\_F (1-\\eta\\_{i-1})^2 \\vf^2\n \\\\\n &\\geq 1- \\ve. \\qedhere\n \\end{split}$$ \nThere is a sequence of constants 1 ≤ *α*0 ≤ *α*1 ≤ ⋯ such that\n \n*α**i*− 1∣*v*∣ ≤ ∣*v*∣*h* ≤ *α**i*∣*v*∣ \nfor all tangent vectors *v* of *S* with foot point in *K**i*, where no index and index *h* indicate measurement with respect to *g* and *h*, respectively. Over *K**i*, the area elements *d**a* of *g* and *d**a**h* of *h* are then estimated by\n \n*α**i*− 2*d**a* ≤ *d**a**h* ≤ *α**i*2*d**a* \nwith corresponding inequalities for the areas of measurable subsets and for integrals of non-negative measurable functions.\nLet now again *φ* be the ground state of *F*, $t\\in\\mathcal R$, and $F\\_t=\\{\\vf^2\\ge t\\}$. In our next result, we estimate the inradius of *F**t* ∩ *K**i* for sufficiently large *i*.\n[cheer] Let *F* be a disc, an annulus, or a cross cap. Assume that *λ*0(*F*) ≤ *θ**λ*0(*S* \\ *K*0) for some 0 < *θ* < 1 and let *δ* > 0. Then there is an integer *i*1 = *i*1(*θ*, *δ*) ≥ 0 such that the inradius *ρ*(*t*) of *F**t* ∩ *K**i* satifies\n \n$$\\coth(\\alpha\\_{i+1}\\rho(t))\n \\le \\frac{2\\alpha\\_{i+1}^3\\sqrt{\\lambda\\_0(F)+ \\delta}}{1- \\delta -t|F \\cap K\\_i|}$$ \nfor all 0 ≤ *t* < (1 − *δ*)/∣*F* ∩ *K**i*∣ and *i* ≥ *i*1.\nIn a first step, we estimate the Rayleigh quotient of $\\eta\\_i\\vf$. Computing as in the proof of, we have\n \n$$\\begin{aligned}\n \\int\\_F |\\nabla(\\eta\\_i\\vf)|^2\n &= \\int\\_F \\nabla(\\eta\\_i^2\\vf)\\nabla\\vf + \\int\\_F|\\nabla\\eta\\_i|^2\\vf^2 \\\\\n &= \\int\\_F \\eta\\_i^2\\vf\\Delta\\vf + \\int\\_F|\\nabla\\eta\\_i|^2\\vf^2 \\\\\n &\\leq \\lambda\\_0(F) \\int\\_F \\eta\\_i^2\\vf^2 + 1/i.\\end{aligned}$$ \nSince *η**i* = 1 on *K**i*, we get $R(\\eta\\_i \\vf) \\le \\lambda\\_0(F) + 2/i$ for all *i* ≥ *i*0(*θ*, 1/2), where *i*0 is taken from. Therefore\n \n$$\\label{coray}\n R(\\eta\\_i \\vf) \\le \\lambda\\_0(F) + \\delta\n \\quad\\text{for all $i \\ge i\\_1(\\theta,\\delta)$,}$$ \nwhere we may assume that *i*1(*θ*, *δ*) ≥ *i*0(*θ*, *δ*). In a second step, we follow the proof of Cheeger’s inequality,. Computing as in, we get\n \n$$\\int\\_F|\\nabla (\\eta\\_i^2 \\vf^2)|\n \\le 2\\sqrt{R(\\eta\\_i\\vf)}\\int\\_F \\eta\\_i^2 \\vf^2\n \\le 2\\sqrt{R(\\eta\\_i\\vf)}.$$ \nBy the coarea formula and and since supp*η**i* ⊆ *K**i* + 1 and *η**i* = 1 on *K**i*, we have\n \n$$\\label{cheeta}\n \\begin{split}\n \\alpha\\_{i+1} \\int\\_F|\\nabla (\\eta\\_i^2\\vf^2)|\n &= \\alpha\\_{i+1} \\int\\_0^\\infty |\\{\\eta\\_i^2\\vf^2=s\\}|\\,ds \\\\\n &\\ge \\int\\_t^\\infty |\\{\\eta\\_i^2\\vf^2=s\\}|\\_h\\,ds \\\\\n &= \\int\\_t^\\infty |\\{\\vf^2=s\\}\\cap K\\_i|\\_h\\,ds \\\\\n &\\hspace{9mm}\n + \\int\\_t^\\infty |\\{\\eta\\_i^2\\vf^2=s\\}\\cap(F\\setminus K\\_i)|\\_h\\,ds\n \\end{split}$$ \nHere we note that, for the integration, it suffices to consider $s\\in\\mathcal R$ which are also regular for $\\eta\\_i^2\\vf^2$. Then $\\{\\vf^2=s\\}$ meets ∂*K**i* transversally and, therefore, $\\{\\eta\\_i^2\\vf^2=s\\}\\cap(F\\setminus K\\_i)$ consists of arcs *a**j* connecting their corresponding end points on $\\{\\vf^2=s\\}\\cap\\partial K\\_i$. Replacing the *a**j* by the corresponding segments *b**j* on ∂*K**i*, we obtain the boundary of the subsurface *F**s* ∩ *K**i*. Now ∣*b**j*∣*h* ≤ ∣*a**j*∣*h* by the choice of the hyperbolic metric *h* on *S* and since the numbers *r**i* defining the *K**i* are positive. Hence we have\n \n$$\\begin{split}\n|\\{\\vf^2=s\\}\\cap K\\_i|\\_h & + |\\{\\eta\\_i^2\\vf^2=s\\} \\cap F\\setminus K\\_i|\\_h\n\\\\\n& = |\\{\\vf^2=s\\}\\cap K\\_i|\\_h + \\sum |a\\_j|\\_h\n\\\\\n& \\geq |\\{\\vf^2=s\\}\\cap K\\_i|\\_h + \\sum |b\\_j|\\_h\n\\\\\n& = |\\partial (\\{\\vf^2 \\geq s\\}\\cap K\\_i)|\\_h,\n\\end{split}$$ \nfor any *s* ∈ R. This implies\n \n$$\\label{cheeta2}\n\\begin{split}\n \\int\\_t^\\infty |\\{\\vf^2=s\\}&\\cap K\\_i|\\_h\\,ds\n + \\int\\_t^\\infty |\\{\\eta\\_i^2\\vf^2=s\\}\\cap(F\\setminus K\\_i)|\\_h\\,ds\n \\\\\n & \\geq \\int\\_t^\\infty |\\partial(\\{\\vf^2\\ge s\\}\\cap K\\_i)|\\_h\\,ds.\n\\end{split}$$ \nFurthermore, by, the connected components of *F**s* ∩ *K**i* are subsurfaces of *F* with piecewise smooth boundary and have non-negative Euler characteristic for any $s\\in\\mathcal R$. Therefore we get\n \n$$\\label{inroh}\n\\begin{split}\n \\int\\_t^\\infty |\\partial(\\{\\vf^2\\ge s\\}&\\cap K\\_i)|\\_h\\,ds \\\\\n &\\ge \\int\\_t^\\infty |\\{\\vf^2\\geq s\\} \\cap K\\_i|\\_h\\coth(\\rho\\_h(s))\\,ds \\\\\n &\\ge \\coth(\\rho\\_h(t)) \\int\\_t^\\infty |\\{\\vf^2 \\geq s\\} \\cap K\\_i|\\_h\\,ds \\\\\n &\\ge \\alpha\\_i^{-2}\\coth(\\alpha\\_i\\rho(t)) \\int\\_t^\\infty |\\{\\vf^2 \\geq s\\} \\cap K\\_i|\\,ds,\n\\end{split}$$ \nby.[c1a],, and. Finally, since *i*1(*θ*, *δ*) ≥ *i*0(*θ*, *δ*),\n \n$$\\label{cheetb}\n\\begin{split}\n \\int\\_t^\\infty |\\{\\vf^2 \\geq s\\} &\\cap K\\_i|\\,ds\n \\ge \\int\\_0^\\infty |\\{\\vf^2 \\geq s\\} \\cap K\\_i|\\,ds - t|F\\cap K\\_i| \\\\\n &= \\int\\_{F\\cap K\\_i}\\vf^2 - t|F\\cap K\\_i|\n \\geq 1- \\delta -t|F\\cap K\\_i|.\n\\end{split}$$ \nfollows now from combining –.\nLemmas [lambdad] and [cheer] will lead to the apriori estimates in and in the proof of, which are essential in the proof of.\nQualititative estimates of Λ(*S*)\n=================================\nIn this section, we prove in the case *χ*(*S*) < 0. Throughout, we let *S* be a complete and connected Riemannian surface of finite type and set\n \n$$\\label{lamdac}\n \\Lambda\\_D(S)= \\inf\\_D \\lambda\\_0(D),\n \\hspace{3mm}\n \\Lambda\\_A(S)= \\inf\\_D \\lambda\\_0(A),\n \\hspace{3mm}\n \\Lambda\\_C(S)= \\inf\\_D \\lambda\\_0(C),$$ \nwhere the infimum is taken over all embedded closed discs *D*, incompressible annuli *A*, and cross caps *C* in $\\mathring S$ with smooth boundary, respectively. As we will explain in, we have\n \nΛ*D*(*S*) ≥ Λ*A*(*S*) and Λ(*S*) = inf{Λ*A*(*S*), Λ*C*(*S*)} \nif the fundamental group of *S* is infinite. Nevertheless, since the case of discs reveals an essential idea of the proof and since we will need the estimate anyway, we include the discussion of Λ*D*(*S*).\nWe fix a hyperbolic metric *h* on *S* as in and denote by *g* the original Riemannian metric of *S*. If not otherwise mentioned, statements refer to *g* and not to *h*.\nWe will use the setup and notation from the previous section. The following assertion is an immediate consequence of.\n[cheet] Let *F* be a closed disc, annulus, or cross cap in *S* and *φ* be the ground state of *F*. Assume that *λ*0(*F*) ≤ *θ**λ*0(*S* \\ *K*0) for some 0 < *θ* < 1. Then the inradius $\\rho(\\ve)$ of $\\{\\vf^2\\ge\\ve\\} \\cap K\\_i$ satisfies $\\rho(\\ve)\\ge\\rho>0$ for all $0<\\ve<1/4|K\\_{i\\_1}|$ and *i* ≥ *i*1 = *i*1(*θ*, 1/2).\nWe now discuss the cases of discs and annuli separately.\n[discs] If *S* is a complete and connected Riemannian surface of finite type with *χ*(*S*) < 0 and $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$, then Λ*D*(*S*) > *λ*0(*S̃*).\nSuppose that there is a sequence of discs *D**n* in *S* with smooth boundary such that $\\lambda\\_0(D\\_n)\\to\\lambda\\_0(\\tilde S)$. Let $\\varphi\\_n\\colon D\\_n\\to\\R$ be the positive *λ*0(*D**n*)-Dirichlet eigenfunction with ∣∣*φ**n*∣∣2 = 1. By passing to a subsequence if necessary, we may assume that *λ*0(*D**n*) ≤ *θ**λ*0(*S* \\ *K*0) for some 0 < *θ* < 1. By and up to passing to a subsequence, there are positive constants $\\ve\\_0$ and *ρ*0 and a point *x*0 ∈ *S* such that *B*(*x*0, 2*ρ*0) is contained in $\\{\\varphi\\_n^2\\ge\\ve\\_0\\} \\cap K$ for all *n*, where *K* = *K**i*1(*θ*, 1/2).\nFix a point $\\tilde x\\_0\\in\\tilde S$ above *x*0. Then there is a unique lift $\\tilde D\\_n$ of *D**n* to $\\tilde S$ containing $\\tilde x\\_0$ such that $\\tilde D\\_n\\to D\\_n$ is a diffeomorphism (including the boundary). Thus we may also lift $\\vf\\_n$ to $\\tilde\\vf\\_n$ on $\\tilde D\\_n$ and extend $\\tilde\\vf\\_n$ to a function $\\tilde\\vf\\_n$ on $\\tilde S\\_n$ by setting $\\tilde\\vf\\_n=0$ on $\\tilde S\\setminus\\tilde D\\_n$. Since the boundary of $\\tilde D\\_n$ is smooth and $\\tilde\\vf\\_n$ is smooth on $\\tilde D\\_n$, it follows that $\\tilde\\vf\\_n\\in H^1\\_0(\\tilde S)$ with *H*1-norm\n \n$$||\\tilde\\vf\\_n||\\_{H^1} = ||\\varphi\\_n||\\_{H^1}^2 = \\lambda\\_0(D\\_n) + 1,$$ \nwhere we use Green’s formula for the second equality. In particular, up to extracting a subsequence, we have weak convergence\n \n$$\\tilde\\vf\\_n \\rightharpoonup \\tilde\\vf\\in H^1\\_0(\\tilde S)\n \\quad\\text{with}\\quad ||\\tilde\\vf||\\_{H^1} \\le \\liminf ||\\varphi\\_n||\\_{H^1}.$$ \nUp to extracting a further subsequence, the sequence of $\\tilde\\vf\\_n$ converges uniformly in any *C**k*-norm in $B(\\tilde x\\_0,\\rho\\_0)$, by Theorem 8.10 in. In particular $\\tilde\\varphi^2\\ge\\ve\\_0$ on $B(\\tilde x\\_0,\\rho\\_0)$.\nBy Theorem 1 of, we may approximate the distance function *d*0 to $\\tilde x\\_0$ in $\\tilde S$ by a smooth function *u* on $\\tilde S$ such that ∣*u* − *d*0∣ ≤ 1 and ∣∇*u*∣ ≤ 2. Then the sublevels *B*(*r*) = {*u* ≤ *r*} form an exhaustion of $\\tilde S$ by compact subsets. Clearly,\n \n$$R(\\tilde\\vf) = \\lim\\_{r\\to\\infty} R(\\tilde\\vf|\\_{B(r)}).$$ \nFurthermore, up to passing to a subsequence, we have weak convergence\n \n$$\\tilde\\vf\\_n|\\_{B(r)}\\rightharpoonup\\tilde\\vf|\\_{B(r)} \\quad\\text{in $H^1(B(r))$}$$ \nand strong convergence\n \n$$\\tilde\\vf\\_n|\\_{B(r)}\\to\\tilde\\vf|\\_{B(r)} \\quad\\text{in $L^2(B(r))$.}$$ \nHence\n \n$$R(\\tilde\\vf|\\_{B(r)})\\le\\liminf R(\\tilde\\vf\\_n|\\_{B(r)}).$$ \nFor any regular value *r* of *u* such that ∂*B*(*r*) intersects $\\partial \\tilde D\\_n$ transversally we have\n \n$$\\begin{split}\n \\int\\_{B(r)}|\\nabla\\tilde\\vf\\_n|^2\n &= \\lambda\\_0(D\\_n) \\int\\_{B(r)} |\\tilde\\vf\\_n|^2\n + \\int\\_{\\partial B(r)}\\tilde\\vf\\_n\\la\\nabla\\tilde\\vf\\_n,\\nu\\ra,\n\\end{split}$$ \nwhere *ν* = ∇*u*/∣∇*u*∣ is the outward unit vector field along ∂*B*(*r*). Clearly, the second term on the right satisfies\n \n$$\\int\\_{\\partial B(r)}\\tilde\\vf\\_n\\la\\nabla\\tilde\\vf\\_n,\\nu\\ra\n \\le \\int\\_{\\partial B(r)} (|\\tilde\\vf\\_n|^2 + |\\nabla\\tilde\\vf\\_n|^2).$$ \nLet now $\\ve>0$ be given. Then there is a sequence of integers *k**m* → ∞ such that, for each *m*, there is a subsequence of *n* → ∞ with\n \n$$\\label{subseq}\n \\int\\_{K(k\\_m)} (|\\tilde\\vf\\_n|^2 + |\\nabla\\tilde\\vf\\_n|^2) < \\ve,$$ \nwhere *K*(*r*) = *B*(*r*) \\ *B*(*r* − 1). If this would not be the case, there would be some $\\ve>0$ and a positive integer *m* such that for all integers *k* ≥ *m* there is an integer *l* such that, for all integers *n* ≥ *l*,\n \n$$\\label{contra}\n\\int\\_{K(k)} (|\\tilde \\varphi\\_n|^2 + |\\nabla \\tilde \\varphi\\_n|^2) \\geq \\ve.$$ \nWe thus find, for any fixed *M*, an index *N* such that holds for *k* = *m*, …, *m* + *M* and *n* ≥ *N*. If we choose *M* such that $M \\ve > \\lambda\\_0(\\tilde S)+1,$ we get a contradiction.\nBy the coarea formula, we have\n \n$$\\int\\_{K(k\\_m)} (|\\tilde \\varphi\\_n|^2 + |\\nabla \\tilde \\varphi\\_n|^2)|\\nabla u|\n = \\int\\_{k\\_m-1}^{k\\_m} \\int\\_{\\partial B(r)} (|\\tilde \\varphi\\_n|^2 + |\\nabla \\tilde \\varphi\\_n|^2).$$ \nSince ∣∇*u*∣ ≤ 2, we then get, for *n* as in, that there is a regular value *r**n* ∈ (*k**m* − 1, *k**m*) of *u* such that\n \n$$\\int\\_{\\partial B(r\\_n)} (|\\tilde\\vf\\_n|^2 + |\\nabla\\tilde\\vf\\_n|^2) < 2\\ve,$$ \nwhere we may also assume that ∂*B*(*r**n*) intersects $\\partial \\tilde D\\_n$ transversally. We obtain\n \n$$\\begin{split}\n R(\\tilde\\vf\\_n|\\_{B(k\\_m)})\n &\\le \\frac{\\int\\_{B(r\\_n)} |\\nabla\\tilde\\vf\\_n|^2}{\\int\\_{B(k\\_m)} |\\tilde\\vf\\_n|^2}\n + \\frac{\\int\\_{K(k\\_m)} |\\nabla\\tilde\\vf\\_n|^2}{\\int\\_{B(k\\_m)}|\\tilde\\vf\\_n|^2} \\\\\n &\\le \\frac{\\int\\_{B(r\\_n)} |\\nabla\\tilde\\vf\\_n|^2}{\\int\\_{B(r\\_n)} |\\tilde\\vf\\_n|^2}\n + \\frac{\\ve}{\\int\\_{B(k\\_m)}|\\tilde\\vf\\_n|^2} \\\\\n &\\le \\lambda\\_0(D\\_n) + \\frac{\n \\int\\_{\\partial B(r\\_n)} \\tilde\\vf\\_n\\la\\nabla\\tilde\\vf\\_n,\\nu\\ra}{\\int\\_{B(r\\_n)} |\\tilde\\vf\\_n|^2}\n + \\frac{\\ve}{\\int\\_{B(k\\_m)}|\\tilde\\vf\\_n|^2}\\\\\n &\\le \\lambda\\_0(D\\_n) + \\frac{2\\ve}{\\int\\_{B(r\\_n)}|\\tilde\\vf\\_n|^2}\n + \\frac{\\ve}{\\int\\_{B(k\\_m)}|\\tilde\\vf\\_n|^2}.\n\\end{split}$$ \nIt follows that\n \n$$R(\\tilde\\vf|\\_{B(k\\_m)})\n \\le \\lambda\\_0(\\tilde S) + \\frac{3\\ve}{\\int\\_{B(\\tilde x\\_0,\\rho\\_0)}|\\tilde\\vf|^2}$$ \nfor all sufficiently large *m*. In conclusion,\n \n$$R(\\tilde\\vf) \\le \\lambda\\_0(\\tilde S).$$ \nSince $\\tilde\\vf\\in H^1\\_0(\\tilde S)$, this implies that $\\tilde\\vf$ is an eigenfunction of the Laplacian with eigenvalue $\\lambda\\_0(\\tilde S)$.\nNow $\\tilde\\vf$ is non-zero on $B(\\tilde x\\_0,\\rho\\_0)$. On the other hand, by the definition of the lifts $\\tilde\\vf\\_n$, $\\tilde\\vf$ vanishes on any other preimages $B(\\tilde x,\\rho\\_0)$ of *B*(*x*0, *ρ*0) under the covering projection $\\tilde S\\to S$. Now the fundamental group of *S* is not trivial, hence there are such preimages $\\tilde x\\in\\tilde S$. Thus we arrive at a contradiction to the unique continuation property for eigenfunctions of Laplacians.\n[annuli] If *S* is a complete and connected Riemannian surface of finite type with *χ*(*S*) < 0 and $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$, then Λ*A*(*S*) > *λ*0(*S̃*).\nTo prove, we assume the contrary and let (*A**n*) be a sequence of incompressible annuli in *S* with smooth boundary such that $\\lambda\\_0(A\\_n)\\to\\lambda\\_0(\\tilde S)$. We may assume that\n \n*λ*0(*A**n*) + 4*δ* < min(Λ*D*(*S*), *θ**λ*0(*S* \\ *K*0)) \nfor all *n* and some fixed constants *δ*, *θ* ∈ (0, 1), by invoking and that $\\lambda\\_0(\\tilde S)<\\lambda\\_0(S\\setminus K\\_0)$ by the choice of *K*0 in. By deforming the *A**n* (slightly), we may also assume that\n \n$\\partial A\\_n$ and $\\partial K\\_i$ intersect transversally \nfor all *n* and *i*. Then the intersections *A**n* ∩ *K**i* and $A\\_n\\cap(S\\setminus\\mathring K\\_i)$ are incompressible subsurfaces of *S* with piecewise smooth boundary.\nRecall the constant *i*1(*θ*, *δ*) from.\n[isotyp] By passing to a subsequence, we may assume that \n[1)]\n[p2] all *A**n* are isotopic in *S* and, for each *i* > *i*1 = *i*1(*θ*, *δ*), exactly one component *A**n*ʹ of *A**n* ∩ *K**i* is an annulus (topologically) isotopic to *A**n*; \n[p4] there is a constant ℓ0 > 0 such that the free homotopy classes of the boundary curves of *A**n* in *A**n* contain curves of length at most ℓ0 with respect to *g* and *h*; \n[p6] there are *x*0 ∈ *S* and $\\rho,\\ve>0$ such that *B*(*x*0, *ρ*) ⊆ *A**n* and such that the ground states $\\vf\\_n$ of *A**n* satisfy $\\varphi\\_n\\ge\\ve$ on *B*(*x*0, *ρ*).\nThe connected components of *A**n* ∩ ∂*K**i* consist of embedded segments connecting two boundary points of *A**n* and of embedded circles in the interior of *A**n*. By the Schoenflies theorem and the topology of *S*, there are the following two possibilities: \n[a)]\n[ca] All connected components of *A**n* ∩ *K**i* are discs. \n[cb] The connected components of *A**n* ∩ *K**i* consist of one annulus *A**n*ʹ (topologically) isotopic to *A**n* and discs. \nNow let *i* > *i*1 = *i*1(*θ*, *δ*). Then $\\eta\\_{i-1}\\vf\\_n$ is a non-zero smooth function on *A**n* with compact support in *A**n* ∩ *K**i* and Rayleigh quotient\n \n$$R(\\eta\\_{i-1}\\vf\\_n) < \\lambda\\_0(A\\_n) + \\delta < \\Lambda\\_D(S),$$ \nby and. Hence, if all the connected components of *A**n* ∩ *K**i* were discs, we would have $R(\\eta\\_{i-1}\\vf\\_n)\\ge\\Lambda\\_D(S)$. Therefore only case [cb]) can occur. Then the Rayleigh quotients of $\\eta\\_{i-1}\\vf\\_n$ on the discs of *A**n* ∩ *K**i* (on which $\\eta\\_{i-1}\\vf\\_n$ does not vanish) must be at least Λ*D*(*S*). Hence the Rayleigh quotient of $\\eta\\_{i-1}\\vf\\_n$ on *A**n*ʹ must be less than *λ*0(*A**n*) + *δ*. In particular, *λ*0(*A**n*ʹ) ≤ *λ*0(*A**n*) + *δ*.\nNote that one of the boundary circles of *A**n*ʹ may only be piecewise smooth. Therefore *A**n*ʹ may only be topologically isotopic to *A**n*.\nNow we let *i* = *i*1 + 1. Since *A**n*ʹ ⊆ *K**i*, we have the uniform area bound ∣*A**n*ʹ∣ ≤ ∣*K**i*∣. This together with the above estimate on *λ*0(*A**n*ʹ) and implies that the length of shortest curves in *A**n*ʹ, which are freely homotopic to the boundary circles of *A**n*ʹ in *A**n*ʹ, is uniformly bounded with respect to *g*. Then their length is also uniformly bounded with respect to *h*, by. In particular, there are only finitely many isotopy types of *A**n*ʹ and, therefore also, of *A**n*. Therefore we may pass to a subsequence so that all of them are isotopic.\nBy and since *K**i* is compact, we may pass to a further subsequence so that all *A**n* ∩ *K**i*, and hence also all *A**n*, contain a geodesic ball *B*(*x*0, *ρ*) such that $\\varphi\\_n\\ge\\ve$ on *B*(*x*0, *ρ*) as claimed.\n[Proof of ] By passing to a subsequence, we may assume that the sequence of *A**n* satisfies all the properties from,, and.\nChoose a shortest (with respect to *h*) closed *h*-geodesic *c* in the free homotopy class in *S* of a generator of the fundamental group of *A**n*. This is possible since the ends of *S* are hyperbolic funnels with respect to *h*. Note that *c* does not depend on *n* since all *A**n* are isotopic. We let $\\hat S$ be the cyclic subcover of $\\tilde S$ to which *c* lifts as a closed $\\hat h$-geodesic $\\hat c$, where $\\hat h$ denotes the lift of *h* to $\\hat S$. Note that all annuli *A**n* are isotopic to a small tubular neighbourhood of *c*. Lifting the corresponding isotopies, we get lifts $\\hat A\\_n\\subseteq\\hat S$ of the annuli *A**n*. Note that $\\hat A\\_n$ is the unique compact component of *π*− 1(*A**n*) and that $\\pi\\colon\\hat A\\_n\\to A\\_n$ is a diffeomorphism which is isometric with respect to *g* and *h* and their respective lifts $\\hat g$ and $\\hat h$.\nDenote by $\\hat x\\_n$ the lift of *x*0 which is contained in $\\hat A\\_n$. If $\\hat x\\_n$ stays at bounded distance to $\\hat c$, the arguments for the case of discs in the proof of apply again and lead to a contradiction since the fundamental group of *S* is not cyclic and $\\lambda\\_0(\\hat S)=\\lambda\\_0(\\tilde S)$.\nSuppose now that $\\hat x\\_n\\to\\infty$ in $\\hat S$. Let $(r,\\theta) \\colon \\hat S \\to \\R\\times(\\R/\\ell\\Z)$ be Fermi coordinates about $\\hat c$, where ℓ denotes the *h*-length of *c* and $\\hat c$, such that $\\hat c=\\{r=0\\}$. Then we have\n \n$$\\hat h = dr^2+\\cosh^2(r)d\\theta^2.$$ \nSince the *h*-length of shortest curves, *c**n*, in the free homotopy class of the boundary curves of *A**n* in *A**n* is bounded by ℓ0, there is a constant *r*0 > 0 such that the lifts $\\hat c\\_n$ of *c**n* to $\\hat A\\_n$ are contained in the region {∣*r*∣ ≤ *r*0} of $\\hat S$.\n The case $\\hat x\\_n \\to \\infty$\nLet $\\hat\\vf\\_n$ be the lift of $\\vf\\_n$ to $\\hat A\\_n$. Then $\\hat\\vf\\_n$ is the ground state of $\\hat A\\_n$ with respect to $\\hat g$ and we have\n \n$$\\label{lowerl2}\n \\int\\_{B(\\hat x\\_n,\\rho)}\\hat\\vf\\_n^2\n = \\int\\_{B(x\\_0,\\rho)} \\vf\\_n^2\n \\ge \\ve^2{\\operatorname{vol}}B(x\\_0,\\rho),$$ \nby.[p6]. Choose *j* > *i*1 = *i*1(*θ*, *δ*) such that\n \n$$\\label{nabet}\n |\\nabla\\eta\\_j|^2 < \\delta\\ve^2{\\operatorname{vol}}B(x\\_0,\\rho).$$ \nSince supp*η**j* ⊆ *K**j* + 1, the *h*-area of supp*η**j* is bounded by the *h*-area ∣*K**j* + 1∣*h* of *K**j* + 1. Now choose an *r*1 > *r*0 such that the $\\hat h$-area of either of the regions − *r*1 ≤ *r* ≤ − *r*0 and *r*0 ≤ *r* ≤ *r*1 in $\\hat S$ is larger than ∣*K**j* + 1∣*h*. Finally, choose a cut off function *χ* on $\\hat S$ such that *χ* = 0 on {∣*r*∣ ≤ *r*1}, *χ* = 1 on {∣*r*∣ ≥ *r*2} for some *r*2 > *r*1, and such that\n \n$$\\label{nabch}\n |\\nabla\\chi|^2 < \\delta\\ve^2{\\operatorname{vol}}B(x\\_0,\\rho).$$ \nComputing as in the proofs of and and with *η* = *η**j* ∘ *π*, we get\n \n$$\\label{upperh1\\_1}\n \\begin{split}\n \\int\\_{\\hat A\\_n } | \\nabla(\\chi\\eta\\hat\\varphi\\_n)|^2\n &= \\int\\_{\\hat A\\_n} \\nabla(\\chi^2\\eta^2\\hat\\vf\\_n)\\nabla\\hat\\vf\\_n\n + \\int\\_{\\hat A\\_n} \\hat\\vf\\_n^2 |\\nabla(\\chi\\eta)|^2\n \\\\\n &= \\lambda\\_0(A\\_n) \\int\\_{\\hat A\\_n} \\chi^2\\eta^2\\hat\\vf\\_n^2\n + \\int\\_{\\hat A\\_n} \\hat\\vf\\_n^2 |\\nabla(\\chi\\eta)|^2\n \\\\\n &\\le \\lambda\\_0(A\\_n) \\int\\_{\\hat A\\_n} \\chi^2\\eta^2\\hat\\vf\\_n^2\n + 4\\delta\\ve^2{\\operatorname{vol}}B(x\\_0,\\rho),\n \\end{split}$$ \nwhere we use and for the passage to the last line.\nSince $\\hat x\\_n\\to\\infty$ and *B*(*x*0, *ρ*) ⊆ *K**j* by the choice of *j*, *χ* = *η* = 1 on $B(\\hat x\\_n,\\rho)$ for all sufficiently large *n*. Combining and, we then get\n \n$$\\label{rayl\\_est}\n R(\\chi\\eta\\hat\\vf\\_n) \\le \\lambda\\_0(A\\_n) + 4\\delta < \\Lambda\\_D(S).$$ \nOn the other hand, ${\\operatorname{supp}}(\\chi\\eta\\hat\\vf\\_n)$ is contained in the lift *B**n* of *A**n* ∩ *K**j* + 1 to $\\hat A\\_n$, intersected with {∣*r*∣ ≥ *r*1}. Now the *h*-area of *B**n* is bounded by ∣*K**j* + 1∣*h* and *B**n* contains *c**n*. Hence *B**n* does not contain loops freely homotopic to *c**n* in the region {∣*r*∣ ≥ *r*1} of $\\hat S$ since otherwise, by the uinqueness of *A**n*ʹ, it would contain one of the regions − *r*1 ≤ *r* ≤ − *r*0 or *r*0 ≤ *r* ≤ *r*1. Hence *B**n* ∩ {∣*r*∣ ≥ *r*1} is a union of discs and, therefore, the Rayleigh quotient of $\\chi\\eta\\hat\\vf\\_n$ has to be at least Λ*D*(*S*), a contradiction to. It follows that the sequence of $\\hat x\\_n$ is bounded.\n[Proof of in the case *χ*(*S*) < 0] By Theorems [discs] and [annuli], we have \n$$\\min\\big(\\Lambda\\_D(S),\\Lambda\\_A(S)\\big) > \\lambda\\_0(\\tilde S)$$ \n for any complete and connected Riemannian surface of finite type with *χ*(*S*) < 0. This implies the assertion of in the case where *S* is orientable since then *S* does not contain cross caps.\nAssume now that *S* is not orientable and let Or(*S*) be the orientation covering space of *S*. Let *C* be a cross cap in *S*. Then the lift of *C* to Or(*S*) is an annulus *A* in Or(*S*) with \n*λ*0(*C*) ≥ *λ*0(*A*) ≥ Λ*A*(Or(*S*)). \n We conclude that $\\Lambda\\_C(S)\\ge\\Lambda\\_A({\\operatorname{Or}}(S))>\\lambda\\_0(\\tilde S)$ as asserted.\nRemarks, examples, and questions\n================================\nIn this section, we collect some loose ends. We start with a comment which gives another argument for calling Λ the analytic systole.\nOn the definition of Λ\n----------------------\nFor complete and connected surfaces *S* with infinite fundamental group, an equivalent definition of the analytic systole is Λ(*S*) = inf*F**λ*0(*F*), where the infimum is taken over incompressible annuli and cross caps *F* with smooth boundary in *S*: \n[a)]\nFor any disc *D* with smooth boundary and any free homotopy class [*c*] of closed curves in *S*, there is an annulus *A* with smooth boundary in *S* containing *D* whose soul belongs to [*c*]. If [*c*] is non-trivial, then *A* is incompressible. Moreover, *λ*0(*A*) ≤ *λ*0(*D*) by the domain monotonicity of *λ*0. \nFor any compressible annulus *A* in *S* with smooth boundary, there is a disc *D* in *S* whose boundary ∂*D* is one of the boundary circles of *A* such that *A* ∪ *D* is a disc in *S* with smooth boundary, by the Schoenflies theorem, and then a) applies. \nCross caps only occur in the case where *S* is not orientable. The soul of a cross cap *C* in *S* is not homotopic to zero in *S*. Since the fundamental group of *S* is torsion free, we get that *C* is incompressible.\nHowever, in view of our previous articles, it is more natural to include discs into the definition. Moreover, it is important in our analysis to handle the case of discs separately.\nOn the essential spectrum\n-------------------------\nThe following result, Proposition 3.6 from formulated for surfaces, is probably folklore. It shows that the essential spectrum of the Laplacian only depends on the geometry of the underlying surface *S* at infinity and that the essential spectrum of the Laplacian is empty if *S* is compact.\n[esp] For a complete Riemannian surface *S* with compact boundary (possibly empty), $\\lambda\\in\\R$ belongs to the essential spectrum of Δ if and only if there is a *Weyl sequence* for *λ*, that is, a sequence $\\vf\\_n$ of smooth functions on *S* with compact support such that \n[1)]\nfor any compact *K* ⊆ *S*, ${\\operatorname{supp}}\\vf\\_n\\cap K=\\emptyset$ for all sufficiently large *n*; \n$\\limsup\\_{n\\rightarrow\\infty}\\|\\vf\\_n\\|\\_2>0$ and $\\lim\\_{n\\rightarrow\\infty}\\|\\Delta\\vf\\_n-\\lambda\\vf\\_n\\|\\_2=0$.\nIn work of Arne Persson and of Richard Froese and Peter Hislop, the bottom of the essential spectrum of Laplacians and more general operators has been characterized in the sense of or, more specifically, in the sense of ; compare with.\n[espb] For a complete Riemannian surface *S*, we have\n \n*λ*ess(*S*) = lim*K**λ*0(*S* \\ *K*), \nwhere *K* runs over the compact subsets of *S*, ordered by inclusion.\n[espc] If *S* is a compact Riemannian surface, then the spectrum of *S* is discrete; that is, *λ*ess(*S*) = ∞.\nSurfaces with cyclic fundamental group\n--------------------------------------\nIn the (unnumbered) lemma on page 551 of, Osserman establishes the following result in the special case of domains in the Euclidean plane.\n[os3] Let *S* be a complete Riemannian surface with boundary (possibly empty) and *p* be a point in the interior of *S*. For sufficiently small $\\ve>0$, let $S\\_\\ve(p)=S\\setminus B\\_\\ve(p)$. Then\n \n$$\\lambda\\_0(S\\_\\ve(p))\\to\\lambda\\_0(S) \\quad\\text{as}\\quad \\ve\\to0.$$ \nThe arguments in also apply to the more general situation of and therefore we skip its proof.\n[equal1] If *S* is a complete Riemannian surface diffeomorphic to sphere or projective plane, then Λ(*S*) = 0.\nFor *p* ∈ *S* and $\\ve>0$, $S\\_\\ve(p)=S\\setminus B\\_\\ve(p)$ is a closed disc or cross cap, respectively, and hence\n \n$$0 \\le \\Lambda(S) \\le \\inf\\_{p,\\ve}\\lambda\\_0(S\\_\\ve(p)) = \\lambda\\_0(S) = 0,$$ \nwhere we use for the penultimate equality.\n[equal3] If S is a complete Riemannian surface diffeomorphic to disc (open or closed), annulus (open, half-open, or closed), or cross cap (open or closed), then *λ*0(*S*) = Λ(*S*).\nIn each case, there exists an increasing sequence of closed discs, annuli, or cross caps *F**n*, respectively, which exhausts the interior $\\mathring S$ of *S*. Hence\n \nΛ(*S*) = lim*λ*0(*F**n*) = *λ*0(*S*), \nby domain monotonicity and the definitions of Λ(*S*) and *λ*0(*S*).\nExamples\n--------\nIt follows from the constructions in that any non-compact and connected surface *S* of finite type carries complete Riemannian metrics of finite or infinite area with discrete spectrum, that is, with *λ*ess(*S*) = ∞. If the fundamental group of *S* is not cyclic, then Λ(*S*) > *λ*0(*S̃*) for any such metric, by. In the following, we extend some constructions from slightly.\nLet $F=\\{(x,y)\\mid x\\ge0,\\, y\\in\\R/L\\Z\\}$ be a funnel with the expanding hyperbolic metric *d**x*2 + cosh(*x*)2*d**y*2. Let $\\kappa\\colon\\R\\to\\R$ be a monotonic smooth function with *κ*(*x*) = − 1 for *x* ≤ 1 and *κ*(*x*) → *κ*∞ ∈ [ − 1, − ∞] as *x* → ∞. Let $j\\colon\\R\\to\\R$ solve *j*ʺ + *κ**j* = 0 with initial condition *j*(0) = 1 and *j*ʹ(0) = 0. Then *j*(*x*) ≥ cosh*x*. The funnel *F* with Riemannian metric *g* = *d**x*2 + *j*(*x*)2*d**y*2 has curvature *K*(*x*, *y*) = *κ*(*x*) ≤ − 1 and infinite area. By comparison, the Rayleigh quotient with respect to *g* of any smooth function $\\vf$ with compact support in the part {*x* ≥ *x*0} of the funnel is at least − *κ*(*x*0)/4.\nLet *S* be a non-compact surface of finite type. Endow *S* with a hyperbolic metric which is expanding along its funnels as above. Replace the hyperbolic metric on the funnels by the above Riemannian metric *g*. Then the new Riemannian metric on *S* is complete with curvature *K* ≤ − 1 and infinite area. By and by what we said above about the Rayleigh quotients, the essential spectrum of the new Riemannian metric is contained in [*κ*∞, ∞). Choosing *κ* such that *κ*∞ is larger than the first Dirichlet eigenvalue *λ*0(*D*) of some smooth closed disc *D* inside the surface yields the estimate\n \n$$\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(D)>\\lambda\\_0(\\tilde S).$$ \nAs a variation, let *j* solve *j*ʺ + *κ**j* = 0 with initial condition *j*(0) = 1 and *j*(∞) = 0. Then *j*ʹ(0) ≤ − 1 and *j*(*x*) ≤ exp( − *x*). The funnel *F* with Riemannian metric *g* = *d**x*2 + *j*(*x*)2*d**y*2 has curvature *K*(*x*, *y*) = *κ*(*x*) and finite area. Again by comparison, the Rayleigh quotient with respect to *g* of any smooth function $\\vf$ with compact support in the part {*x* ≥ *x*0} of the funnel is at least − *κ*(*x*0)/4.\nLet *S* be a non-compact surface of finite type, and choose *r* > 0 such that coth(*r*) = − *j*ʹ(0). Now *S* minus the parts {*x* ≥ *r*} of its funnels carries hyperbolic metrics which are equal to *d**x*2 + *j*0(*x*)2*d**y*2 along the parts {*x* < *r*} of its funnels, where *j*0(*x*) = sinh(*r* − *x*)/sinh(*r*). Then *j*0(*x*) = *j*(*x*) for *x* < min{1, *r*}. Hence any such hyperbolic metric, restricted to *S* minus the parts {*x* ≥ min{1, *r*}} of its funnels, when combined with *g* along the funnels, defines a smooth and complete Riemannian metric on *S* which has curvature *K* ≤ − 1 and finite area. Choosing *κ* and *D* as in the first case, we again have $\\lambda\\_{{\\operatorname{ess}}}(S)>\\lambda\\_0(\\tilde S)$.\nGeneric metrics\n---------------\nIn view of and we are now prepared for the proof of.\n[Proof of ] For the first part observe that for any complete Riemannian surface *S* of finite type, we have\n \n$$\\lambda\\_0(S) \\le \\lambda\\_0(\\tilde S) \\le \\Lambda(S),$$ \nby and, respectively. We conclude that\n \n$$\\lambda\\_0(S) = \\lambda\\_0(\\tilde S) = \\Lambda(S)$$ \nfor the surfaces considered in Propositions [equal1] and [equal3]. These surfaces are precisely the ones with cyclic fundamental group.\nThe second part follows immediately from, so we are only left with the proof of the third part.\nLet *S* be a non-compact surface of finite type and *g* be a complete Riemannian metric on *S* with\n \n$$\\lambda\\_0(\\tilde S,\\tilde g)=\\lambda\\_{\\rm ess}(S,g).$$ \nBy Theorem [main1] we then have\n \n$$\\lambda\\_0(\\tilde S,\\tilde g)=\\Lambda(S,g)=\\lambda\\_{\\rm ess}(S,g).$$ \nNow assume that $\\lambda\\_{\\rm ess}(S, g)>0$. For *n* ≥ 1, let *F**n* ⊆ *S* be a smooth closed disc, annulus or cross cap with\n \n$$\\lambda\\_0(F\\_n,g)\n < e^{1/n+1}\\Lambda(S,g)\n = e^{1/n+1} \\lambda\\_{\\rm ess}(S,g).$$ \nChoose exhaustions of *S* by compact subsets *K**n* and *L**n* and smooth functions *h**n* such that, for all *n* ≥ 1,\n \n$$F\\_n \\subseteq \\mathring K\\_n \\subseteq K\\_n \\subseteq \\mathring L\\_n$$ \nand\n \n$$e^{-1/n}\\le h\\_n\\le 1, \\quad\n \\text{$h\\_n=1$ on $K\\_n$}, \\quad\n \\text{$h\\_n=e^{-1/n}$ on $S\\setminus L\\_n$.}$$ \nThere exists a smooth function *f* = *f**t* = *f*(*t*, *x*) on (0, 1] × *S* with *f*1/*n* = *h**n* such that, for all 0 < *t* ≤ 1/*n*,\n \n$$\\text{$f\\_t=1$ on $K\\_n$\n and $f\\_t=e^{-t}$ on $S\\setminus L\\_n$}$$ \nand such that *f* is monotonically decreasing in *t*. Since *f**t* = 1 on *K**n* for *t* ≤ 1/*n* and the *K**n* exhaust *S*, *f* can be smoothly extended to [0, 1] × *S* by setting *f*0 = 1.\nLet *g**t* = *f**t**g*. Then *g**t* is a smooth family of conformal metrics on *S* and is a continuous curve of metrics with respect to the uniform distance. For *t* ≤ 1/*n*, we have\n \n$$\\Lambda(S,g\\_t) \\le \\lambda\\_0(F\\_n,g\\_t) =\\lambda\\_0(F\\_n,g)\n < e^{1/n+1}\\lambda\\_{\\rm ess}(S,g).$$ \nSince the Dirichlet integral is invariant under conformal change in dimension two, we obtain, for 1/*n* + 1 ≤ *t* ≤ 1/*n*,\n \n$$\\Lambda(S,g\\_t)\n < e^{1/n+1}\\lambda\\_{\\rm ess}(S,g)\n \\le e^t \\lambda\\_{\\rm ess}(S,g)\n = \\lambda\\_{\\rm ess}(S,g\\_t).$$ \nInvoking Theorem [main1] we conclude that for all *t* > 0 one has the inequality:\n \n$$\\lambda\\_0(\\tilde S,\\tilde g\\_t)\n < \\Lambda(S,g\\_t)\n < \\lambda\\_{\\rm ess}(S,g\\_t).$$ \nIt remains to show that the set of metrics *g* on *S* that satisfy the strict inequality\n \nΛ(*S*, *g*) > *λ*0(*S̃*, *g̃*) \nis an open set in the uniform *C*∞ topology. This follows from the fact that two metrics that are close to each other in the uniform *C*∞ topology are quasi-isometric by a quasi-isometry with quasi-isometry constant close to 1.\n[remsgeneric] 1) Any two complete Riemannian metrics *g*0, *g*1 on *S* of finite uniform distance are quasi-isometric. This implies, that there is a constant *C* > 0 depending on the distance of *g*0 to *g*1 such that\n \n$$C^{-1}\\lambda\\_{\\rm ess}(S,g\\_0) \\leq \\lambda\\_{\\rm ess}(S,g\\_1) \\leq C \\lambda\\_{\\rm ess}(S,g\\_0).$$ \nIn particular, we have that\n* $\\lambda\\_{\\rm ess}(S,g\\_0)$ is finite iff $\\lambda\\_{\\rm ess}(S,g\\_1)$ is finite,\n* $\\lambda\\_{\\rm ess}(S,g\\_0) = 0 $ iff $\\lambda\\_{\\rm ess}(S,g\\_1)=0.$\n2) The above construction can be extended to get metrics with\n \n$$\\lambda\\_{\\rm ess}(S,g\\_t)=e^t\\lambda\\_{\\rm ess}(S,g)$$ \nfor all *t* ≥ 0. \n3) If *S* is non-compact, any complete hyperbolic metric on *S* satisfies $\\lambda\\_0(\\tilde S,\\tilde g)=\\lambda\\_{\\rm ess}(S,g)=1/4.$ \n4) If *S* is non-compact, any complete Riemannian metric on *S*, which is in zeroth order asymptotic to a flat cylinder $\\R/L\\Z \\times [0,\\infty),$ has $\\lambda\\_0(\\tilde{S},g)=\\lambda\\_{\\rm ess}(S,g)=0.$\nProblems and questions\n----------------------\nWe collect some questions naturally arising in view of our results. Let *S* be a compact and connected Riemannian surface with negative Euler characteristic. \n[1)]\n[opde](Optimal design) For a given compact subsurface *T* of $\\mathring S$ with smooth boundary ∂*T* ≠ ∅, we may consider the constant \nΛ*T*(*S*) = inf*F**λ*0(*F*), \n where *F* runs over all subsurfaces of *S* isotopic to *T*. The analytic systole is an infimum over such constants. It is interesting to ask for estimates of Λ*T*(*S*). The infimum is probably achieved by degenerate *F*, where ∂*F* is mapped onto a graph Γ in *S* such that *S* \\ Γ is diffeomorphic to the interior of *T*. In fact, for any *F* isotopic to *T*, there is a graph Γ in *S* such that *F* ⊆ *S* \\ Γ and such that *S* \\ Γ is isotopic to the interior of *F*. Hence, by domain monotonicity, Λ*T*(*S*) is the infimum over all *λ*0(*S* \\ Γ), where Γ runs through such graphs. What are the optimal graphs? This circle of problems is related to the work of Helffer, Hoffman-Ostenhof, and Terracini.\n[rigi](Rigidity) The inequality *λ*− *χ*(*S*)(*S*) > Λ(*S*), mentioned in the introduction, raises the question whether there is another natural geometric constant Λʹ(*S*) > Λ(*S*), where we only have the weak inequality *λ*− *χ*(*S*) ≥ Λʹ(*S*) and where equality occurs only for a distinguished class of Riemannian metrics.\n[rigi1](Another rigidity) The last part of suggests that hyperbolic metrics on non-compact surfaces of finite type are among a small collection of metrics that satisfy\n \nΛ(*S*) = *λ*0(*S̃*) = *λ**e**s**s*(*S*). \nIt would be interesting to see what other implications this equality has on the metric. If we rescale the metric by a function *f*: *S* → (0, 1] which is 1 outside a compact set, then $\\lambda\\_0(\\tilde S)$ can only increase, while $\\lambda\\_{\\rm ess}(S)$ remains unaffected. Using our main theorem we can see that the new metric also satisfy the above equality. Hence one can not have a rigidity among all smooth metrics. Also, observe that points 1) and 4) of imply that there is no such rigidity for metrics with $\\lambda\\_{\\rm ess}(S)=0.$\n[higherdim](Higher dimensions) All our definitions extend in a natural way to higher dimensional manifolds. For instance, we may define the analytic systole of an *n*-dimensional manifold *M* by Λ(*M*) = infΩ*λ*0(Ω), where Ω runs over all tubular neighborhoods about essential simple loops in *M*. By, we have $\\Lambda(M)\\ge\\lambda\\_0(\\tilde M)$. One may ask whether the strict inequality holds true under reasonable assumptions on *M*. Our methods seem to be too weak to adress this question.\nOn *λ*0 under coverings\n=======================\nIn, Brooks states that, for a Riemannian covering $\\pi\\colon\\hat M\\to M$ of complete Riemannian manifolds without boundary, the bottom of the spectrum remains unchanged provided the covering is normal with amenable covering group and that *M* has finite topological type, that is, that *M* is the union of finitely many simplices. We use the corresponding result in the case where the covering is normal with cyclic fundamental group, but where the boundaries of $\\hat M$ and *M* may not be empty. In fact, in we also claim that the results there remain true for Schrödinger operators Δ + *V* with non-negative potential *V*.\nBy the proof of or, the bottom *λ*0(*M*, *V*) of the spectrum of a Schrödinger operator Δ + *V* on a complete and connected Riemannian manifold *M* with boundary (possibly empty) with non-negative potential *V*. is the top of the positive spectrum of Δ + *V*. Now for a Riemannian covering $\\pi\\colon\\hat M\\to M$ of complete and connected Riemannian manifolds with boundary (possibly empty) and non-negative potentials *V* and $\\hat V=V\\circ\\pi$, the lift of a positive *λ*-eigenfunction of Δ + *V* on *M* to $\\hat M$ is a positive *λ*-eigenfunction of $\\Delta+\\hat V$. Therefore\n \n$$\\label{pomv}\n \\lambda\\_0(M,V) \\le \\lambda\\_0(\\hat M,\\hat V)$$ \nin this situation. Since the lift of a square integrable function on *M* to $\\hat M$ is square integrable if the covering is finite, the reverse inequality holds for such coverings, but does not hold in general.\n[brooks2] Let $\\pi\\colon\\hat M\\to M$ be a normal Riemannian covering of complete and connected Riemannian manifolds with boundary (possibly empty) with infinite cyclic covering group. Let $V\\colon M\\to\\R$ be a smooth non-negative function and set $\\hat V=V\\circ\\pi$. Then $\\lambda\\_0(M,V)=\\lambda\\_0(\\hat M,\\hat V)$.\nThe case of the standard Laplacian corresponds to the case *V* = 0. Note that we do not need to assume that *M* has finite topological type in the sense of Brooks.\n[Proof of ] By, we have $\\lambda\\_0(M,V)\\le\\lambda\\_0(\\hat M,\\hat V)$. To show the reverse inequality, let $\\ve>0$ and $\\vf$ be a smooth function on *M* with compact support in the interior of *M* and Rayleigh quotient\n \n$$R(\\vf) = \\int\\_M \\{ |\\nabla\\vf|^2 + V\\vf^2 \\} \\big/ \\int\\_M\\vf^2 < \\lambda\\_0(M,V)+\\ve.$$ \nThe strategy is now to cut off the lift $\\hat\\vf=\\vf\\circ\\pi$ of $\\vf$ to $\\hat M$ conveniently so that the Rayleigh quotient of the new function is bounded by $R(\\vf)+\\ve$.\nWe note first that the covering $\\pi\\_0\\colon\\R\\to\\R/\\Z$ is universal. Hence the covering *π* is the pull back of *π*0 by a smooth map $f\\colon M\\to\\R/\\Z$. Without loss of generality, we may assume that $[0]\\in\\R/\\Z$ is a regular value of *f*. Then *f*− 1([0]) is a smooth hypersurface of *M*.\nUp to covering transformation, there is a unique lift $\\hat f\\colon\\hat M\\to\\R$ of *f*. Then $\\pi^{-1}(f^{-1}([0]))\\subseteq \\hat M$ is the union of the smooth hypersurfaces $\\hat f^{-1}(k)$, $k\\in\\Z$. Moreover, $\\hat f^{-1}([k,k+1])$ is a smooth fundamental domain for the action of $\\Z$ on $\\hat M$, for all $k\\in\\Z$, and ${\\operatorname{supp}}\\hat\\vf\\cap\\hat f^{-1}([a,b])$ is compact, for all *a* ≤ *b*.\nLet *η*0 be a non-negative smooth function on $\\hat M$ which is positive on $\\hat f^{-1}([0,1])$ and which has support in $\\hat f^{-1}([-1,2])$. Set *η**k* = *η*0 ∘ *λ**k*, where *λ**k* denotes the action of $k\\in\\Z$ on $\\hat M$, and $\\zeta\\_k=\\eta\\_k/\\sum\\_{j\\in\\Z}\\eta\\_j$. Note that the sum in the denominator on the right is well defined since it is locally finite. Then (*ζ**k*) is a partition of unity on $\\hat M$ such that *ζ**k* = *ζ*0 ∘ *λ**k*. In particular, since *ζ*0 has support in $\\hat f^{-1}([-1,2])$ and ${\\operatorname{supp}}\\hat\\vf\\cap\\hat f^{-1}([-1,2])$ is compact, there is a uniform bound ∣∇*ζ**k*∣ ≤ *C* on ${\\operatorname{supp}}\\hat\\vf$. Therefore\n \n*χ**k* = ∑− 1 ≤ *j* ≤ *k* + 1*ζ**j* \nis a smooth cut-off function on $\\hat M$ with values in [0, 1] which is equal to 1 on *f*− 1([0, *k*]), has support in *f*− 1([ − 2, *k* + 2]), and gradient bounded by 3*C* on ${\\operatorname{supp}}\\hat\\vf$. We conclude that\n \n$$\\int\\_{\\hat M} (\\chi\\_k\\hat\\vf)^2 \\ge k \\int\\_{M} \\vf^2,\n \\quad\n \\int\\_{\\hat M} \\hat V(\\chi\\_k\\hat\\vf)^2 \\le (k+4) \\int\\_M V\\vf^2,$$ \nand, using Young’s inequality,\n \n$$\\begin{aligned}\n \\int\\_{\\hat M} |\\nabla(\\chi\\_k\\hat\\vf)|^2\n &\\le (1+\\delta) \\int\\_{\\hat M} |\\nabla\\vf|^2 + \\left(1+\\frac{1}{\\delta} \\right)\\int\\_{{\\operatorname{supp}}\\nabla \\chi\\_k } |\\nabla\\chi\\_k|^2\\hat\\vf^2 \\\\\n &\\le (1+\\delta)(k+4) \\int\\_M |\\nabla\\vf|^2 + \\left(1+\\frac{1}{\\delta} \\right)36C^2 \\int\\_M \\vf^2,\\end{aligned}$$ \nsince ${\\operatorname{supp}}\\nabla \\chi\\_k \\subseteq \\hat f^{-1}([-2,0]\\cup[k,k+2]).$ Choosing *δ* small enough, we hence find that $\\chi\\_k\\hat\\vf$ is a smooth function on $\\hat M$ with compact support such that $R(\\chi\\_k\\hat\\vf)<R(\\vf)+\\ve$ for all sufficiently large *k*.\nHMZ2\nN.Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. *J.Math.Pures Appl.* (9) **36** (1957), 235–249.\nD.Azagra, J.Ferrera, F.López-Mesas, Y.Rangel, Smooth approximation of Lipschitz functions on Riemannian manifolds. *J.Math.Anal.Appl.* **326** (2007), no. 2, 1370–1378.\nW. 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With applications to Schrödinger operators.* Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. x+337 pp.\nM.G.Katz, *Systolic geometry and topology.* With an appendix by Jake P. Solomon. Mathematical Surveys and Monographs, 137. Amer. Math. Society, Providence, RI, 2007. xiv+222 pp.\nH.Matthiesen, *The theorem of Otal and Rosas, *λ*2*g* − 2 > 1/4.* Bachelor Thesis, University of Bonn, 2013. 29 pp.\nS.Mondal, Systole and *λ*2*g* − 2 of closed hyperbolic surfaces of genus *g*. *Enseign. Math.* **60** (2014) no. 2, 1-23.\nR.Osserman, A note on Hayman’s theorem on the bass note of a drum. *Comment. Math. Helv.* **52** (1977), no. 4, 545–555.\nJ.-P.Otal and E.Rosas, Pour toute surface hyperbolique de genre *g*, *λ*2*g* − 2 > 1/4. *Duke Math. J.* **150** (2009), no. 1, 101–115.\nP.Schmutz, Small eigenvalues on Riemann surfaces of genus 2. *Invent. Math.* **106** (1991), no. 1, 121–138.\nD.Sullivan, Related aspects of positivity in Riemannian geometry. *J. Differential Geom.* **25** (1987), no. 3, 327–351.\nK.Uhlenbeck, Generic properties of eigenfunctions. *Amer. J. Math.* **98** (1976), no. 4, 1059–1078.\n---\n1. *Acknowledgments.* We would like to thank the Max Planck Institute for Mathematics in Bonn, the Hausdorff Center for Mathematics in Bonn, and Indiana University at Bloomington for their support and hospitality. We would also like to thank the referees whose criticism led to a better presentation in the paper.[↩](#fnref"},"data_id":{"kind":"string","value":"arxiv_0182772"}}},{"rowIdx":45694,"cells":{"content":{"kind":"string","value":"Fast and Compact Self-Stabilizing Verification, Computation,and Fault Detection of an MST\n=========================================================================================\nThis paper demonstrates the usefulness of distributed local verification of proofs, as a tool for the design of self-stabilizing algorithms. In particular, it introduces a somewhat generalized notion of distributed local proofs, and utilizes it for improving the time complexity significantly, while maintaining space optimality. As a result, we show that optimizing the memory size carries at most a small cost in terms of time, in the context of Minimum Spanning Tree (MST). That is, we present algorithms that are both time and space efficient for both constructing an MST and for verifying it. This involves several parts that may be considered contributions in themselves.\nFirst, we generalize the notion of local proofs, trading off the time complexity for memory efficiency. This adds a dimension to the study of distributed local proofs, which has been gaining attention recently. Specifically, we design a (self-stabilizing) proof labeling scheme which is memory optimal (i.e., *O*(log*n*) bits per node), and whose time complexity is *O*(log2*n*) in synchronous networks, or *O*(Δlog3*n*) time in asynchronous ones, where Δ is the maximum degree of nodes. This answers an open problem posed by Awerbuch and Varghese (FOCS 1991). We also show that Ω(log*n*) time is necessary, even in synchronous networks. Another property is that if *f* faults occurred, then, within the required detection time above, they are detected by some node in the *O*(*f*log*n*) locality of each of the faults.\nSecond, we show how to enhance a known transformer that makes input/output algorithms self-stabilizing. It now takes as input an efficient construction algorithm and an efficient self-stabilizing proof labeling scheme, and produces an efficient self-stabilizing algorithm. When used for MST, the transformer produces a memory optimal self-stabilizing algorithm, whose time complexity, namely, *O*(*n*), is significantly better even than that of previous algorithms. (The time complexity of previous MST algorithms that used Ω(log2*n*) memory bits per node was *O*(*n*2), and the time for optimal space algorithms was *O*(*n*∣*E*∣).) Inherited from our proof labelling scheme, our self-stabilising MST construction algorithm also has the following two properties: (1) if faults occur after the construction ended, then they are detected by some nodes within *O*(log2*n*) time in synchronous networks, or within *O*(Δlog3*n*) time in asynchronous ones, and (2) if *f* faults occurred, then, within the required detection time above, they are detected within the *O*(*f*log*n*) locality of each of the faults. We also show how to improve the above two properties, at the expense of some increase in the memory.\nDistributed network algorithms, Locality, Proof labels, Minimum spanning tree, Distributed property verification, Self-stabilization, Fast fault detection, Local fault detection.\nIntroduction\n============\nMotivation\n----------\nIn a non-distributed context, solving a problem is believed to be, sometimes, much harder than verifying it (e.g., for NP-Hard problems). Given a graph *G* and a subgraph *H* of *G*, a task introduced by Tarjan is to check whether *H* is a Minimum Spanning Tree (MST) of *G*. This non-distributed verification seems to be just slightly easier than the non-distributed computation of an MST. In the distributed context, the given subgraph *H* is assumed to be represented distributively, such that each node stores pointers to (some of) its incident edges in *H*. The verification task consists of checking whether the collection of pointed edges indeed forms an MST, and if not, then it is required that at least one node *raises an alarm*. It was shown recently that such an MST verification task requires the same amount of time as the MST computation . On the other hand, assuming that each node can store some information, i.e., *a label*, that can be used for the verification, the time complexity of an MST verification can be as small as 1, when using labels of size Θ(log2*n*) bits per node, where *n* denotes the number of nodes. To make such a proof labeling scheme a useful algorithmic tool, one needs to present a *marker* algorithm for computing those labels. One of the contributions of the current paper is a time and memory efficient marker algorithm.\nEvery decidable graph property (not just an MST) can be verified in a short time given large enough labels . A second contribution of this paper is a generalization of such schemes to allow a reduction in the memory requirements, by trading off the locality (or the time). In the context of MST, yet another (third) contribution is a reduced space proof labeling scheme for MST. It uses just *O*(log*n*) bits of memory per node (asymptotically the same as the amount of bits needed for merely representing distributively the MST). This is below the lower bound of Ω(log2*n*) of. The reason this is possible is that the verification time is increased to *O*(log2*n*) in synchronous networks and to *O*(Δlog3*n*) in asynchronous ones, where Δ is the maximum degree of nodes. Another important property of the new scheme is that any fault is detected rather close to the node where it occurred. Interestingly, it turns out that a logarithmic time penalty for verification is unavoidable. That is, we show that Ω(log*n*) time for an MST verification scheme is necessary if the memory size is restricted to *O*(log*n*) bits, even in synchronous networks. (This, by the way, means that a verification with *O*(log*n*) bits, cannot be silent, in the sense of ; this is why they could not be of the kind introduced in ).\nGiven a long enough time, one can verify *T* by recomputing the MST. An open problem posed by Awerbuch and Varghese is to find a synchronous MST verification algorithm whose time complexity is smaller than the MST computation time, yet with a small memory. This problem was introduced in in the context of self-stabilization, where the verification algorithm is combined with a non-stabilizing *construction* protocol to produce a stabilizing protocol. Essentially, for such purposes, the verification algorithm repeatedly checks the output of the non-stabilizing construction protocol, and runs the construction algorithm again if a fault at some node is detected. Hence, the construction algorithm and the corresponding verification algorithm are assumed to be designed together. This, in turn, may significantly simplify the checking process, since the construction algorithm may produce output variables (labels) on which the verification algorithm can later rely. In this context, the above mentioned third contribution solves this open problem by showing an *O*(log2*n*) time penalty (in synchronous networks) when using optimal *O*(log*n*) memory size for the MST verification algorithm. In contrast, if we study MST *construction* instead of MST verification, time lower bounds which are polynomial in *n* for MST construction follow from (even for constant diameter graphs).\nOne known application of some methods of distributed verification is for general transformers that transform non-self-stabilizing algorithms to self-stabilizing ones. The fourth contribution of this paper is an adaptation of the transformer of such that it can transform algorithms in our context. That is, while the transformer of requires that the size of the network and its diameter are known, the adapted one allows networks of unknown size and diameter. Also, here, the verification method is a proof labeling scheme whose verifier part is self-stabilizing. Based on the strength of the original transformer of (and that of the companion paper it uses), our adaptation yields a result that is rather useful even without plugging in the new verification scheme. This is demonstrated by plugging in the proof labeling schemes of, yielding an algorithm which already improves the time of previous *O*(log2*n*) memory self-stabilizing MST construction algorithm, and also detects faults using 1 time and at distance at most *f* from each fault (if *f* faults occurred).\nFinally, we obtain an optimal *O*(log*n*) memory size, *O*(*n*) time asynchronous self-stabilizing MST construction algorithm. The state of the art time bound for such optimal memory algorithms was *O*(*n*∣*E*∣). In fact, our time bound improves significantly even the best time bound for algorithms using polylogarithmic memory, which was *O*(*n*2).\nMoreover, our self-stabilizing MST algorithm inherits two important properties from our verification scheme, which are: (1) the time it takes to detect faults is small: *O*(log2*n*) time in a synchronous network, or *O*(Δlog3*n*) in an asynchronous one; and (2) if some *f* faults occur, then each fault is detected within its *O*(*f*log*n*) neighbourhood. Intuitively, a short detection distance and a small detection time may be helpful for the design of local correction, for fault confinement, and for fault containment algorithms. Those notions were introduced to combat the phenomena of faults “spreading” and “contaminating” non-faulty nodes. For example, the infamous crash of the ARPANET (the predecessor of the Internet) was caused by a fault in a single node. This caused old updates to be adopted by other nodes, who then generated wrong updates affecting others. This is an example of those non-faulty nodes being contaminated. The requirement of *containment* is that such a contamination does not occur, or, at least, that it is contained in a small radius around the faults. The requirement of *confinement* allows the contamination of a state of a node, as long as this contamination is not reflected in the output (or the externally visible actions) of the non-faulty nodes. Intuitively, if the detection distance is short, non-faulty nodes can detect the faults and avoid being contaminated.\nRelated work\n------------\nThe distributed construction of an MST has yielded techniques and insights that were used in the study of many other problems of distributed network protocols. It has also become a standard to check a new paradigm in distributed algorithms theory. The first distributed algorithm was proposed by, its complexity was not analyzed. The seminal paper of Gallager, Humblet, and Spira presented a message optimal algorithm that used *O*(*n*log*n*) time, improved by Awerbuch to *O*(*n*) time, and later improved in to $O(D + \\sqrt{n} \\log^\\* n)$, where *D* is the diameter of the network. This was coupled with an almost matching lower bound of $\\Omega(D+\\sqrt{n})$.\nProof labeling schemes were introduced in. The model described therein assumes that the verification is restricted to 1 unit of time. In particular, a 1 time MST verification scheme was described there using *O*(log2*n*) bits per node. This was shown to be optimal in. In, Göös and Suomela extend the notion of proof labeling schemes by allowing constant time verification, and exhibit some efficient proof labeling schemes for recognizing several natural graph families. In all these schemes, the criterion to decide failure of a proof (that is, the detection of a fault) is the case that at least one node does not manage to verify (that is, detects a fault). The global state passes a proof successfully if all the nodes verify successfully. This criterion for detection (or for a failure to prove) was suggested by in the contexts of self stabilization, and used in self stabilization (e.g. ) as well as in other other contexts.\nSelf-stabilization deals with algorithms that must cope with faults that are rather severe, though of a type that does occur in reality. The faults may cause the states of different nodes to be inconsistent with each other. For example, the collection of marked edges may not be an MST.\nTable [tab:alg-compare] summarizes the known complexity results for self stabilizing MST construction algorithms. The first several entrees show the results of using (to generate an MST algorithm automatically) the known transformer of Katz and Perry, that extends automatically non self stabilizing algorithms to become self stabilizing. The transformer of Katz and Perry assumes a leader whose memory must hold a snapshot of the whole network. The time of the resulting self-stabilizing MST algorithm is *O*(*n*) and the memory size is *O*(∣*E*∣*n*). We have attributed a higher time to in the table, since we wanted to remove its assumption of a known leader, to make a fair comparison to the later papers who do not rely on this assumption.\nTo remove the assumption, in the first entry we assumed the usage of the only leader election known at the time of. That is, in, the first self-stabilizing leader election algorithm was proposed in order to remove the assumptions of that a leader and a spanning tree are given. The combination of and implied a self-stabilizing MST in *O*(*n*2) time. (Independently, a leader election algorithm was also presented by ; however, we cannot use it here since it needed an extra assumption that a bound on *n* was known; also, its higher time complexity would have driven the complexity of the transformed MST algorithm higher than the *O*(*n*2) stated above.)\nUsing unbounded space, the time of self-stabilizing leader election was later improved even to *O*(*D*) (the actual diameter). The bounded memory algorithms of or, together with and, yield a self-stabilizing MST algorithm using *O*(*n*∣*E*∣log*n*) bits per node and time *O*(*D*log*n*) or *O*(*n*).\n[tab:alg-compare]\n|p3.5cm|c|c|c|p5cm| `Algorithm` & space & time & Asynch & comment \n& & & & \n$\\rm \\cite{katz-perry} \\rm $++& *O*(∣*E*∣*n*) & *O*(*n*2)& yes & \n& & & & \n$\\rm \\cite{katz-perry} \\rm $+ + & unbounded & *O*(*D*) & yes & The 2nd component can be replaced by, assuming the $\\cal{LOCAL}$ model. \n& & & & \n$\\rm \\cite{katz-perry}$+ $\\rm \\cite{awerbuch-mst} \\rm $+ $\\rm \\cite{awerbuch-kutten-patt-et-al}\\rm $ & *O*(∣*E*∣*n*)log*n* & *O*(min{*D*log*n*, *n*})& yes& The third component here can be replaced by $\\rm \\cite{afek-bremler}\\rm $ or by $\\rm \\cite{datta2008} \\rm $. \n& & & & \n$\\rm \\cite{gupta-srimani-see-sebastian}\\rm $ & *O*(*n*log*n*) & *O*(*n*) & no & Implies an *O*(*n*2) time bound in asynchronous networks, assuming a good bound on the network size is known. The time is based on assuming the $\\cal{LOCAL}$ model. \n& & & & \n$\\rm \\cite{higham}\\rm $ & *O*(log*n*) & Ω(∣*E*∣*n*) & yes & The time complexity is based on the assumption that a good bound on the network diameter is known. \n& & & & \n$\\rm \\cite{sebastian-disc2009} \\rm $ & *O*(log*n*) & Ω(∣*E*∣*n*) & yes & Aims to exchanging less bits with neighbours than $\\rm \\cite{higham}\\rm $. Assumes a leader is known. \n& & & & \n$\\rm \\cite{sebastian-disc-2010} \\rm $ & *O*(log2(*n*)) & *O*(*n*2) & yes & \n& & & & \nCurrent paper & *O*(log*n*) & *O*(*n*) & yes & \nAntonoiu and Srimani presented a self stabilizing algorithm whose complexities were not analyzed. As mentioned by, the model in that paper can be transformed to the models of the other papers surveyed here, at a high translation costs. Hence, the complexities of the algorithm of may grow even further when stated in these models. Gupta and Srimani presented an *O*(*n*log*n*) bits algorithm. Higham and Liang improved the core memory requirement to *O*(log*n*), however, the time complexity went up again to Ω(*n*∣*E*∣). An algorithm with a similar time complexity and a similar memory per node was also presented by Blin, Potop-Butucaru, Rovedakis, and Tixeuil. This latter algorithm exchanges less bits with neighbours than does the algorithm of. The algorithm of addressed also another goal- even during stabilization it is *loop free*. That is, it also maintains a tree at all times (after reaching an initial tree). This algorithm assumes the existence of a unique leader in the network (while the algorithm in the current paper does not). However, this does not seem to affect the order of magnitude of the time complexity.\nNote that the memory size in the last two algorithms above is the same as in the current paper. However, their time complexity is *O*(∣*E*∣*n*) versus *O*(*n*) in the current paper. The time complexity of the algorithm of Blin, Dolev, Potop-Butucaru, and Rovedakis improved the time complexity of to *O*(*n*2) but at the cost of growing the memory usage to *O*(log2*n*). This may be the first paper using labeling schemes for the design of a self-stabilizing MST protocol, as well as the first paper implementing the algorithm by Gallager, Humblet, and Spira in a self-stabilizing manner without using a general transformer.\nAdditional studies about MST verification in various models appeared in. In particular, Kor et al. shows that the verification from scratch (without labels) of an MST requires $\\Omega(\\sqrt{n}+D)$ time and Ω(∣*E*∣) messages, and that these bounds are tight up to poly-logarithmic factors. We note that the memory complexity was not considered in, and indeed the memory used therein is much higher than the one used in the current paper. The time lower bound proof in was later extended in to apply for a variety of verification and computation tasks.\nThis paper has results concerning distributed verification. Various additional papers dealing with verification have appeared recently, the models of some of them are rather different than the model here. Verification in the $\\cal{LOCAL}$ model (where congestion is abstracted away) was studied in from a computational complexity perspective. That paper presents various complexity classes, shows separation between them, and provides complete problems for these classes. In particular, the class NLD defined therein exhibits similarities to the notion of proof labeling schemes. Perhaps the main result in is a sharp threshold for the impact of randomization on local decision of hereditary languages. Following that paper, showed that the threshold in holds also for any non-hereditary language, assuming it is defined on path topologies. In addition, showed further limitations of randomness, by presenting a hierarchy of languages, ranging from deterministic, on the one side of the spectrum, to almost complete randomness, on the other side. Still, in the $\\cal{LOCAL}$ model, studied the impact of assuming unique identifiers on local decision. We stress that the memory complexity was not considered in neither nor in its follow-up papers.\nOur results\n-----------\nThis paper contains the following two main results.\n#### (1) Solving an open problem posed by Awerbuch and Varghese :\nIn the context of self-stabilization, an open problem posed in is to find a (synchronous) MST verification algorithm whose time complexity is smaller than the MST computation time, yet with a small memory. Our first main result solves this question positively by constructing a time efficient self-stabilizing verification algorithm for an MST while using optimal memory size, that is *O*(log*n*) bits of memory per node. More specifically, the verification scheme takes as input a distributed structure claimed to be an MST. If the distributed structure is indeed an MST, and if a marker algorithm properly marked the nodes to allow the verification, and if no faults occur, then our algorithm outputs *accept* continuously in every node. However, if faults occur (including the case that the structure is not, in fact, an MST, or that the marker did not perform correctly), then our algorithm outputs *reject* in at least one node. This *reject* is outputted in time *O*(log2*n*) after the faults cease, in a synchronous network. (Recall, lower bounds which are polynomial in *n* for MST *construction* are known even for synchronous networks.) In asynchronous networks, the time complexity of our verification scheme grows to *O*(Δlog3*n*). We also show that Ω(log*n*) time is necessary if the memory size is restricted to *O*(log*n*), even in synchronous networks. Another property of our verification scheme is that if *f* faults occurred, then, within the required detection time above, they are detected by some node in the *O*(*f*log*n*) locality of each of the faults. Moreover, we present a distributed implementation of the marker algorithm whose time complexity for assigning the labels is *O*(*n*), under the same memory size constraint of *O*(log*n*) memory bits per node.\n#### (2) Constructing an asynchronous self-stabilizing MST construction algorithm which uses optimal memory (*O*(log*n*) bits) and runs in *O*(*n*) time:\nIn our second main result, we show how to enhance a known transformer that makes input/output algorithms self-stabilizing. It now takes as input an efficient construction algorithm and an efficient self-stabilizing proof labeling scheme, and produces an efficient self-stabilizing algorithm. When used with our verification scheme, the transformer produces a memory optimal self-stabilizing MST construction algorithm, whose time complexity, namely, *O*(*n*), is significantly better even than that of previous algorithms. (Recall, the time complexity of previous MST algorithms that used Ω(log2*n*) memory bits per node was *O*(*n*2), and the time for optimal space algorithms was *O*(*n*∣*E*∣).) Inherited from our verification scheme, our self-stabilising MST construction algorithm also has the following two properties. First, if faults occur after the construction ended, then they are detected by some nodes within *O*(log2*n*) time in synchronous networks, or within *O*(Δlog3*n*) time in asynchronous ones, and second, if *f* faults occurred, then, within the required detection time above, they are detected within the *O*(*f*log*n*) locality of each of the faults. We also show how to improve these two properties, at the expense of some increase in the memory.\nOutline\n-------\nPreliminaries and some examples of simple, yet useful, proof labeling schemes are given in Section [sec:preliminaries]. An intuition is given in Section [sec:intuition]. A building block is then given in Section [sec:mst-construction]. Namely, that section describes a synchronous MST construction algorithm in *O*(log*n*) bits memory size and *O*(*n*) time. Section [sec:section5] describes the construction of parts of the labeling scheme. Those are the parts that use labeling schemes of the kind described in - namely, schemes that can be verified in one time unit. These parts use the MST construction (of Section [sec:mst-construction]) to assign the labels. Sections [sec:proof], [sec:viewing], and [sub:utilizing] describe the remaining part of the labeling scheme. This part is a labeling scheme by itself, but of a new kind. It saves on memory by distributing information. Specifically, Section [sec:proof] describes how the labels should look if they are constructed correctly (and if an MST is indeed represented in the graph). The verifications, in the special case that no further faults occur, is described in Section [sec:viewing]. This module verifies (alas, not in constant time) by moving the distributed information around, for a “distributed viewing”. Because the verification is not done in one time unit, it needs to be made self stabilizing. This is done in Section [sub:utilizing]. Section [sec:lower] presents a lower bound for the time of a proof labeling scheme for MST that uses only logarithmic memory. (Essentially, the proof behind this lower bound is based on a simple reduction, using the rather complex lower bound given in.)\nThe efficient self-stabilizing MST algorithm is given in Section [sec:full-mst-alg]. Using known transformers, we combine efficient MST verification schemes and (non-self-stabilizing) MST construction schemes to yield efficient self-stabilizing schemes. The MST construction algorithm described in Section [sec:mst-construction] is a variant of some known time efficient MST construction algorithms. We show there how those can also be made memory efficient (at the time, this complexity measure was not considered), and hence can be used as modules for our optimal memory self-stabilizing MST algorithm.\nPreliminaries\n=============\n[sec:def]\nSome general definitions\n------------------------\nWe use rather standard definitions; a reader unfamiliar with these notions may refer to the model descriptions in the rich literature on these subjects. In particular, we use rather standard definitions of self-stabilization (see, e.g. ). Note that the assumptions we make below on time and time complexity imply (in self stabilization jargon) a distributed daemon with a very strong fairness. When we speak of asynchronous networks, this implies a rather fine granularity of atomicity. Note that the common self stabilization definitions include the definitions of faults. We also use standard definitions of graph theory (including an edge weighted graph *G* = (*V*, *E*), with weights that are polynomial in *n* = ∣*V*∣) to represent a network (see, e.g. ). Each node *v* has a unique identity `ID`(*v*) encoded using *O*(log*n*) bits. For convenience, we assume that each adjacent edge of each node *v* has some label that is unique at *v* (edges at different nodes may have the same labels). This label, called a *port-number*, is known only to *v* and is independent of the port-number of the same edge at the other endpoint of the edge. (Clearly, each port-number can be assumed to be encoded using *O*(log*n*) bits). Moreover, the network can store an object such as an MST (Minimum Spanning Tree) by having each node store its *component* of the representation. A component *c*(*v*) at a node *v* includes a collection of pointers (or port-numbers) to neighbours of *v*, and the collection of the components of all nodes induces a subgraph *H*(*G*) (an edge is included in *H*(*G*) if and only if at least one of its end-nodes points at the other end-node). In the verification scheme considered in this current paper, *H*(*G*) is supposed to be an MST and for simplicity, we assume that the component of each node contains a single pointer (to the parent, if that node is not defined as the root). It is not difficult to extend our verification scheme to hold also for the case where each component can contain several pointers. Note that the definitions in this paragraph imply a lower bound of Ω(log*n*) bits on the memory required at each node to even represent an MST (in graphs with nodes of high degree).\nSome additional standard () parts of the model include the assumption that the edge weights are distinct. As noted often, having distinct edge weights simplifies our arguments since it guarantees the uniqueness of the MST. Yet, this assumption is not essential. Alternatively, in case the graph is not guaranteed to have distinct edge weights, we may modify the weights locally as was done in. The resulted modified weight function *ω*ʹ(*e*) not only assigns distinct edge weights, but also satisfies the property that the given subgraph *H*(*G*) is an MST of *G* under *ω*( ⋅ ) if and only if *H*(*G*) is an MST of *G* under *ω*ʹ( ⋅ ).[4](#fn4)\nWe use the (rather common) ideal time complexity which assumes that a node reads all of its neighbours in at most one time unit, see e.g. . Our results translate easily to an alternative, stricter, *contention* time complexity, where a node can access only one neighbour in one time unit. The time cost of such a translation is at most a multiplicative factor of Δ, the maximum degree of a node (it is not assumed that Δ is known to nodes).\nAs is commonly assumed in the case of self-stabilization, each node has only some bounded number of memory bits available to be used. Here, this amount of memory is *O*(log*n*).\nUsing protocols designed for message passing\n--------------------------------------------\nWe use a self stabilizing transformer of Awerbuch and Varghese as a building block. That protocol was designed for the message passing model. Rather than modifying that transformer to work on the model used here (which would be very easy, but would take space), we use emulation. That is, we claim that any self stabilizing protocol designed for the model of (including the above transformer) can be performed in the model used here, adapted from. This is easy to show: simply use the current model to implement the links of the model of. To send a message from node *v* to its neighbour *u*, have *v* write its shared variable that (only *v* and) *u* can read. This value can be read by *u* after one time unit in a synchronous network as required from a message arrival in the model of. Hence, this is enough for synchronous networks.\nIn an asynchronous network, we need to work harder to simulate the sending and the receiving of a message, but only slightly harder, given known art. Specifically, in an asynchronous network, an event occurs at *u* when this message arrives. Without some additional precaution on our side, *u* could have read this value many times (per one writing) resulting in duplications: multiple message “arriving” while we want to emulate just one message. This is solved by a self stabilizing data link protocol, such as the one used by, since this is also the case in a data link protocol in message passing systems where a link may loose a package. There, a message is sent repeatedly, possibly many times, until an acknowledgement from the receiver tells the sender that the message arrived. The data link protocol overcomes the danger of duplications by simply numbering the messages modulo some small number. That is, the first message is sent repeatedly with an attached “sequence number” zero, until the first acknowledgement arrives. All the repetitions of the second message have as attachments the sequence number 1, etc. The receiver then takes just one of the copies of the first message, one of the copies of the second, etc. A self stabilized implementation of this idea in a shared memory model appears in using (basically, to play the role of the sequence number) an additional shared variable called the “toggle”, which can take one of three values.[5](#fn5) When *u* reads that the toggle of *v* changes, *u* can emulate the arrival of a message. In terms of time complexity, this protocol takes a constant time, and hence sending (an emulated) message still takes a constant time (in terms of complexity only) as required to emulate the notion of ideal time complexity of. Note that the memory is not increased.\nWave&Echo\n---------\nWe use the well known Wave&Echo (PIF) tool. For details, the readers are referred to. For completeness, we remind the reader of the overview of Wave&Echo when performed over a rooted tree. It is started by the tree root, and every node who receives the wave message forwards it to its children. The wave can carry a command for the nodes. A leaf receiving the wave, computes the command, and sends the output to its parent. This is called an *echo*. A parent, all of whose children echoed, computes the command itself (possibly using the outputs sent by the children) and then sends the echo (with its own output) to its parent. The Wave&Echo terminates at the root when all the children of the root echoed, and when the root executed the command too.\nIn this paper, the Wave&Echo activations carry various commands. Let us describe first two of these commands, so that they will also help clarify the notion of Wave&Echo and its application. The first example is the command to sum up values residing at the nodes. The echo of a leaf includes its value. The echo of a parent includes the sum of its own value and the sums sent by its children. Another example is the counting of the nodes. This is the same as the sum operation above, except that the initial value at a node is 1. Similarly to summing up, the operation performed by the wave can also be a logical `OR`.\nProof labeling schemes\n----------------------\nIn, the authors consider a framework for maintaining a distributed proof that the network satisfies some given predicate Ψ, e.g., that *H*(*G*) is an MST. We are given a predicate Ψ and a graph family $\\cF$ (in this paper, if Ψ and $\\cF$ are omitted, then Ψ is MST and $\\cF$ (or $\\cF(n)$) is all connected undirected weighted graphs with *n* nodes). A *proof labeling scheme* (also referred to as a *verification* algorithm) includes the following two components.\n* A *marker* algorithm $\\cM$ that generates a label $\\cM(v)$ for every node *v* in every graph $G\\in \\cF$.\n* A *verifier*, that is a distributed algorithm $\\cV$, initiated at each node of a *labeled* graph $G\\in \\cF$, i.e., a graph whose nodes *v* have labels *L*(*v*) (not necessarily correct labels assigned by a marker). The verifier at each node is initiated separately, and at an arbitrary time, and runs forever. The verifier may *raise an alarm* at some node *v* by outputting “no” at *v*.\nIntuitively, if the verifier at *v* raises an alarm, then it detected a fault. That is, for any graph $G\\in \\cF$,\n* If *G* satisfies the predicate Ψ and if the label at each node *v* is $\\cM(v)$ (i.e., the label assigned to *v* by the marker algorithm $\\cM)$ then no node raises an alarm. In this case, we say that the verifier *accepts* the labels.\n* If *G* does not satisfy the predicate Ψ, then for *any* assignment of labels to the nodes of *G*, after some finite time *t*, there exists a node *v* that raises an alarm. In this case, we say that the verifier *rejects* the labels.\nNote that the first property above concerns only the labels *produced by the marker algorithm* $\\cM$, while the second must hold even if the labels are assigned by some adversary. We evaluate a proof labeling scheme $(\\cM,\\cV)$ by the following complexity measures.\n* The *memory size*: the maximum number of bits stored in the memory of a single node *v*, taken over all the nodes *v* in all graphs $G\\in \\cF(n)$ that satisfy the predicate Ψ (and over all the executions); this includes: (1) the bits used for encoding the identity `ID`(*v*), (2) the marker memory: number of bits used for constructing and encoding the labels, and (3) the verifier memory: the number of bits used during the operation of the verifier[6](#fn6).\n* The (ideal) *detection time*: the maximum, taken over all the graphs $G\\in \\cF(n)$ that do *not* satisfy the predicate Ψ and over all the labels given to nodes of *G* by adversaries (and over all the executions), of the time *t* required for some node to raise an alarm. (The time is counted from the *starting time*, when the verifier has been initiated at all the nodes.)\n* The *detection distance*: for a faulty node *v*, this is the (hop) distance to the closest node *u* raising an alarm within the detection time after the fault occurs. The detection distance of the scheme is the maximum, taken over all the graphs having at most *f* faults, and over all the faulty nodes *v* (and over all the executions), of the detection distance of *v*.\n* The (ideal) *construction-time*: the maximum, taken over all the graphs $G\\in \\cF(n)$ that satisfy the predicate Ψ (and over all the executions), of the time required for the marker $\\cM$ to assign labels to all nodes of *G*. Unless mentioned otherwise, we measure construction-time in synchronous networks only.\nIn our terms, the definitions of allowed only detection time complexity 1. Because of that, the verifier of at a node *v*, could only consult the neighbours of *v*. Whenever we use such a scheme, we refer to it as a *1-proof labeling scheme*, to emphasis its running time. Note that a 1-proof labelling scheme is trivially self-stabilzying. (In some sense, this is because they “silently stabilize”, and “snap stabilize”.) Also, in, if *f* faults occurred, then the detection distance was *f*.\nGeneralizing the complexities to a computation\n----------------------------------------------\nIn Section [sec:proof-labeling], we defined the memory size, detection time and the detection distance complexities of a *verification* algorithm. When considering a (self-stabilizing) computation algorithm, we extend the notion of the memory size to include also the bits needed for encoding the component *c*(*v*) at each node. Recall, the definition of a component *c*(*v*) in general, and the special case of *c*(*v*) for MST, are given in Section [sec:preliminaries]. (Recall, from Section [sec:proof-labeling], that the size of the component was excluded from the definition of memory size for verification because, there, the designer of the verification scheme has no control over the nodes’ components.)\nThe notions of detection time and the detection distance can be carried to the very common class of self-stabilizing *computation* algorithms that use fault detection. (Examples for such algorithms are algorithms that have silent stabilization.) Informally, algorithms in this class first compute an output. After that, all the nodes are required to stay in some *output state* where they (1) output the computation result forever (unless a fault occurs); and (2) check repeatedly until they discover a fault. In such a case, they recompute and enter an output state again. Let us term such algorithms *detection based* self-stabilizing algorithms. We define the *detection time* for such algorithms to be the time from a fault until the detection. However, we only define the detection time (and the detection distance) for faults that occur after all the nodes are in their output states. (Intuitively, in the other cases, stabilization has not been reached yet anyhow.) The *detection distance* is the distance from a node where a fault occurred to the closest node that detected a fault.\nSome examples of 1-proof labeling schemes\n-----------------------------------------\nAs a warm-up exercise, let us begin by describing several simple 1-proof labeling schemes, which will be useful later in this paper. The first two examples are taken from and are explained there in more details. The reader familiar with proof labeling schemes may skip this subsection. \nLet *f**s**p**a**n* denote the predicate such that for any input graph *G* satisfies, *f**s**p**a**n*(*G*) = 1 if *H*(*G*) is a spanning tree of *G*, and *f**s**p**a**n*(*G*) = 0 otherwise. We now describe an efficient 1-proof labeling scheme $(\\cM,\\cV)$ for the predicate *f**s**p**a**n* and the family of all graphs. Let us first describe the marker $\\cM$ operating on a “correct instance”, i.e., a graph *G* where *T* = *H*(*G*) is indeed a spanning tree of *G*. If there exists a node *u* whose component does not encode a pointer to any of its adjacent edges (observe that there can be at most one such node), we root *T* at *u*. Otherwise, there must be two nodes *v* and *w* whose components point at each other. In this case, we root *T* arbitrarily at either *v* or *w*. Note that after rooting *T*, the component at each non-root node *v* points at *v*’s parent. The label given by $\\cM$ to a node *v* is composed of two parts. The first part encodes the identity `ID`(*r*) of the root *r* of *T*, and the second part of the label of *v* encodes the distance (assuming all weights of edges are 1) between *v* and *r* in *T*.\nGiven a labeled graph, the verifier $\\cV$ operates at a node *v* as follows: first, it checks that the first part of the label of *v* agrees with the first part of the labels of *v*’s neighbours, i.e., that *v* agrees with its neighbours on the identity of the root. Second, let *d*(*v*) denote the number encoded in the second part of *v*’s label. If *d*(*v*) = 0 then $\\cV$ verifies that `ID`(*v*) = `ID`(*r*) (recall that `ID`(*r*) is encoded in the first part of *v*’s label). Otherwise, if *d*(*v*) ≠ 0 then the verifier checks that *d*(*v*) = *d*(*u*) + 1, where *u* is the node pointed at by the component at *v*. If at least one of these conditions fails, the verifier $\\cV$ raises an alarm at *v*. It is easy to get convinced that $(\\cM,\\cV)$ is indeed a 1-proof labeling scheme for the predicate *f**s**p**a**n* with memory size *O*(log*n*) and construction time *O*(*n*).\n#### Remark.\nObserve that in case *T* = *H*(*G*) is indeed a (rooted) spanning tree of *G*, we can easily let each node *v* know which of its neighbours in *G* are its children in *T* and which is its parent. Moreover, this can be done using one unit of time and label size *O*(log*n*) bits. To see this, for each node *v*, we simply add to its label its identity `ID`(*v*) and the identity `ID`(*u*) of its parent *u*. The verifier at *v* first verifies that `ID`(*v*) is indeed encoded in the right place of its label. It then looks at the label of its parent *u*, and checks that *v*’s candidate for being the identity of *u* is indeed `ID`(*u*). Assume now that these two conditions are satisfied at each node. Then, to identify a child *w* in *T*, node *v* should only look at the labels of its neighbours in *G* and see which of them encoded `ID`(*v*) in the designated place of its label. \nDenote by *f**s**i**z**e* the boolean predicate such that *f**s**i**z**e*(*G*) = 1 if and only if one designated part of the label *L*(*v*) at each node *v* encodes the number of nodes in *G* (informally, when *f**s**i**z**e* is satisfied, we say that each node ‘knows’ the number of nodes in *G*). Let us denote the part of the label of *v* that is supposed to hold this number by *L*ʹ(*v*).\nIn, the authors give a 1-proof labeling scheme $(\\cM,\\cV)$ for *f**s**i**z**e* with memory size *O*(log*n*). The idea behind their scheme is simple. First, the verifier checks that *L*ʹ(*u*) = *L*ʹ(*v*) for every two adjacent nodes *u* and *v* (if this holds at each node then all nodes must hold the same candidate for being the number of nodes). Second, the marker constructs a spanning tree *T* rooted at some node *r* (and verifies that this is indeed a spanning tree using the Example $\\SP$ above). Third, the number of nodes in *T* is aggregated upwards along *T* towards its root *r*, by keeping at the label $\\cM(v)$ of each node *v*, the number of nodes *n*(*v*) in the subtree of *T* hanging down from *v*. This again is easily verified by checking at each node *v* that *n*(*v*) = 1 + ∑*u* ∈ `Child`(*v*)*n*(*u*), where `Child`(*v*) is the set of children of *v*. Finally, the root verifies that *n*(*r*) = *L*ʹ(*r*). It is straightforward that $(\\cM,\\cV)$ is indeed a 1-proof labeling scheme for the predicate *f**s**i**z**e* with memory size *O*(log*n*) and construction time *O*(*n*). \nConsider a tree *T* rooted at *r*, and let *h* denote the height of *T*. Denote by *f**h**e**i**g**h**t* the boolean predicate such that *f**h**e**i**g**h**t*(*T*) = 1 if and only if one designated part of the label *L*(*v*) at each node encodes the same value *x*, where *h* ≤ *x* (informally, when *f**h**e**i**g**h**t* is satisfied, we say that each node ‘knows’ an upper bound of 2*x* on the diameter *D* of *T*). Let us denote the part of the label of *v* that is supposed to hold this number by *L*ʹ(*v*). We sketch a simple 1-proof labeling scheme $(\\cM,\\cV)$ for *f**h**e**i**g**h**t*. First, the verifier checks that *L*ʹ(*u*) = *L*ʹ(*v*) for every two adjacent nodes *u* and *v* (if this holds at each node then all nodes must hold the same value *x*). Second, similarly to the proof labeling scheme for *f**s**p**a**n* given in Example $\\SP$ above, the label in each node *v* contains the distance *d*(*v*) in the tree from *v* to the root. Each non-root node verifies that the distance written at its parent is one less than the distance written at itself, and the root verifies that the distance written at itself is 0. Finally, each node *v* verifies also that *x* ≥ *d*(*v*). If no node raises an alarm then *x* is an upper bound on the height. On the other hand, if the value *x* is the same at all nodes and *x* is an upper bound on the height then no node raises an alarm. Hence the scheme is a 1-proof labeling scheme for the predicate *f**h**e**i**g**h**t* with memory size *O*(log*n*) and construction time *O*(*n*).\nOverview of the MST verification scheme and the intuition behind it\n===================================================================\nThe final MST construction algorithm makes use of several modules. The main technical contribution of this paper is the module that verifies that the collection of nodes’ components is indeed an MST. This module in itself is composed of multiple modules. Some of those, we think may be useful by themselves. To help the reader avoid getting lost in the descriptions of all the various modules, we present, in this section, an overview of the MST verification part.\nGiven a graph *G* and a subgraph that is represented distributively at the nodes of *G*, we wish to produce a self-stabilizing proof labeling scheme that verifies whether the subgraph is an MST of *G*. By first employing the (self-stabilizing) 1-proof labeling scheme mentioned in Example $\\SP$, we may assume that the subgraph is a rooted spanning tree of *G* (otherwise, at least one node would raise an alarm). Hence, from now on, we focus on a spanning tree *T* = (*V*(*G*), *E*(*T*)) of a weighted graph *G* = (*V*(*G*), *E*(*G*)), rooted at some node *r*(*T*), and aim at verifying the minimality of *T*.\nBackground and difficulties\n---------------------------\nFrom a high-level perspective, the proof labeling scheme proves that *T* could have been computed by an algorithm that is similar to that of GHS, the algorithm of Gallager, Humblet, and Spira’s described in. This leads to a simple idea: when *T* is a tree computed by such an algorithm, *T* is an MST. Let us first recall a few terms from. A *fragment* *F* denotes a connected subgraph of *T* (we simply refer it to a *subtree*). Given a fragment *F*, an edge (*v*, *u*) ∈ *E*(*G*) whose one endpoint *v* is in *F*, while the other endpoint *u* is not, is called *outgoing* from *F*. Such an edge of minimum weight is called a *minimum outgoing* edge from *F*. A fragment containing a single node is called a *singleton*. Recall that GHS starts when each node is a fragment by itself. Essentially, fragments merge over their minimum outgoing edges to form larger fragments. That is, each node belongs to one fragment *F*1, then to a larger fragment *F*2 that contains *F*1, etc. This is repeated until one fragment spans the network. A tree constructed in that way is an MST. Note that in GHS, the collection of fragments is a laminar family, that is, for any two fragments *F* and *F*ʹ in the collection, if *F* ∩ *F*ʹ ≠ ∅ then either *F* ⊆ *F*ʹ or *F*ʹ ⊆ *F* (see, e.g. ). Moreover, each fragment has a *level*; in the case of *v* above, *F*2’s level is higher than that of *F*1. This organizes the fragments in a *hierarchy* $\\cH$, which is a tree whose nodes are fragments, where fragment *F*1 is a descendant in $\\cH$ of *F*2 if *F*2 contains *F*1. GHS manages to ensure that each node belongs to at most one fragment at each level, and that the total number of levels is *O*(log*n*). Hence, the hierarchy $\\cH$ has *O*(log*n*) height.\nThe marker algorithm in our proof labeling scheme performs, in some sense, a reverse operation. If *T* is an MST, the marker “slices” it back into fragments. Then, the proof labeling scheme computes for each node *v*:\nThe (unique) name of each of the fragments *F**j* that *v* belongs to,\nthe level of *F**j*, and\nthe weight of *F**j*’s minimum outgoing edge.\nNote that each node participates in *O*(log*n*) fragments, and the above “piece of information” per fragment requires *O*(log*n*) bits. Hence, this is really too much information to store in one node. As we shall see later, the verification scheme distributes this information and then brings it to the node without violating the memory size bound *O*(log*n*). For now, it suffices to know that given these pieces of information and the corresponding pieces of information of their neighbours, the nodes can verify that *T* could have been constructed by an algorithm similar to GHS. That way, they verify that *T* is an MST. Indeed, the 1-proof labeling schemes for MST verification given in follow this idea employing memory size of *O*(log2*n*) bits. (There, each node keeps *O*(log*n*) pieces, each of *O*(log*n*) bits.)\nThe current paper allows each node to hold only *O*(log*n*) memory bits. Hence, a node has room for only a constant number of such pieces of information at a time. One immediate idea is to store some of *v*’s pieces in some other nodes. Whenever *v* needs a piece, some algorithm should move it towards *v*. Moving pieces would cost time, hence, realizing some time versus memory size trade-off.\nUnfortunately, the total (over all the nodes) number of pieces in the schemes of is Ω(*n*log*n*). Any way one would assign these pieces to nodes would result in the memory of some single node needing to store Ω(log*n*) pieces, and hence, Ω(log2*n*) bits. Thus, one technical step we used here is to reduce the total number of pieces to *O*(*n*), so that we could store at each node just a constant number of such pieces. However, each node still needs to use Ω(log*n*) pieces. That is, some pieces may be required by many nodes. Thus, we needed to solve also a combinatorial problem: locate each piece “close” to each of the nodes needing it, while storing only a constant number of pieces per node.\nThe solution of this combinatorial problem would have sufficed to construct the desired scheme in the $\\cal{LOCAL}$ model . There, node *v* can “see” the storage of nearby nodes. However, in the congestion aware model, one actually needs to move pieces from node to node, while not violating the *O*(log*n*) memory per node constraint. This is difficult, since, at the same time *v* needs to see its own pieces, other nodes need to see their own ones.\nA very high level overview\n--------------------------\nGoing back to GHS, one may notice that its correctness follows from the combination of two properties:\n**P1. (Well-Forming)** The existence of a hierarchy tree $\\cH$ of fragments, satisfying the following:\nEach fragment $F\\in \\cH$ has a unique *selected* outgoing edge (except when *F* is the whole tree *T*).\nA (non-singleton) fragment is obtained by merging its children fragments in $\\cH$ through their selected outgoing edges.\n**P2. (Minimality)** The selected outgoing edge of each fragment is its minimal outgoing edge.\nIn our proof labeling scheme, we verify the aforementioned two properties separately. In Section [sec:section5], we show how to verify the first property, namely, property Well-Forming. This turns out to be a much easier task than verifying the second property. Indeed, the Well-Forming property can be verified using a 1-proof labeling scheme, while still obeying the *O*(log*n*) bits per node memory constraint. Moreover, the techniques we use for verifying the Well-Forming are rather similar to the ones described in. The more difficult verification task, namely, verifying the Minimality property P2, is described in Section [sec:proof]. This verification scheme requires us to come up with several new techniques which may be considered as contributions by themselves. We now describe the intuition behind these verifications.\nVerifying the Well-Forming property (described in detail in Sections [sec:mst-construction] and [sec:section5])\n---------------------------------------------------------------------------------------------------------------\nWe want to show that *T* could have been produced by an algorithm similar to GHS. Crucially, since we care about the memory size, we had to come up with a new MST construction algorithm that is similar to GHS but uses only *O*(log*n*) memory bits per node and runs in time *O*(*n*). This MST construction algorithm, called `SYNC\\_MST`, can be considered as a synchronous variant of GHS and is described in Section [sec:mst-construction].\nIntuitively, for a correct instance (the case *T* is an MST), the marker algorithm $\\cM$ produces a hierarchy of fragments $\\cH$ by following the new MST construction algorithm described in Section [sec:mst-construction]. Let ℓ = *O*(log*n*) be the height of $\\cH$. For a fixed level *j* ∈ [0, ℓ], it is easy to represent the partition of the tree into fragments of level *j* using just one bit per node. That is, the root *r*ʹ of each fragment *F*ʹ of level *j* has 1 in this bit, while the nodes in *F*ʹ \\ {*r*ʹ} have 0 in this bit. Note, these nodes in *F*ʹ \\ {*r*ʹ} are the nodes below *r*ʹ (further away from the root of *T*), until (and not including) reaching additional nodes whose corresponding bit is 1. Hence, to represent the whole hierarchy, it is enough to attach a string of length ℓ + 1-bits at each node *v*. The string at a node *v* indicates, for each level *j* ∈ [0, ℓ], whether *v* is the root of the fragment of level *j* containing *v* (if one exists).\nNext, still for correct instances, we would like to represent the selected outgoing edges distributively. That is, each node *v* should be able to detect, for each fragment *F* containing *v*, whether *v* is an endpoint of the selected edge of *F*. If *v* is, it should also know which of *v*’s edges is the selected edge. This representation is used later for verifying the two items of the Well-Forming property specified above. For this purpose, first, we add another string of ℓ + 1 entries at each node *v*, one entry per level *j*. This entry specifies, first, whether there exists a level *j* fragment *F**j*(*v*) containing *v*. If *F**j*(*v*) does exist, the entry specifies whether *v* is incident to the corresponding selected edge. Note, storing the information at *v* specifying the pointers to all the selected edges of the fragments containing it, may require *O*(log2*n*) bits of memory at *v*. This is because there may be *O*(log*n*) fragments containing *v*; each of those may select an edge at *v* leading to an arbitrary neighbour of *v* in the tree *T*; if *v* has many neighbours, each edge may cost *O*(log*n*) bits to encode. The trick is to distribute the information regarding *v*’s selected edges among *v*’s children in *T*. (Recall that *v* can look at the data structures of *v*’s children.)\nThe strings mentioned in the previous paragraphs are supposed to define a hierarchy and selected outgoing edges from the fragments of the hierarchy. However, on incorrect instances, if corrupted, the strings may not represent the required. For example, the strings may represent more than one selected edge for some fragment. Hence, we need also to attach proof labels for verifying the hierarchy and the corresponding selected edges represented by those strings. Fortunately, for proving the Well-Forming property only, it is not required to verify that the represented hierarchy (and the corresponding selected edges) actually follow the MST construction algorithm (which is the case for correct instances). In Section [sec:section5], we present 1-proof labeling schemes to show that the above strings represent *some* hierarchy with corresponding selected edges, and that the Well-Forming property does hold for that hierarchy.\nVerifying The Minimality property (described in detail in Sections [sec:proof], [sub:3.2] and [sub:utilizing])\n--------------------------------------------------------------------------------------------------------------\nA crucial point in the scheme is letting each node *v* know, for each of its incident edges (*v*, *u*) ∈ *E* and for each level *j*, whether *u* and *v* share the same level *j* fragment. Intuitively, this is needed in order to identify outgoing edges. For that purpose, we assign each fragment a unique identifier, and *v* compares the identifier of its own level *j* fragment to the identifier of *u*’s level *j* fragment.\nConsider the number of bits required to represent the identifiers of all the fragments that a node *v* participates in. There exists a method to assign unique identifiers such that this total number is only *O*(log*n*). Unfortunately, we did not manage to use that method here. Indeed, there exists a marker algorithm that assigns identifiers according to that method. However, we could not find a low space and short time method for the verifier to verify that the given identifiers of the fragments were indeed assigned in that way. (In particular, we could not verify efficiently that the given identifiers are indeed unique).\nHence, we assign identifiers according to another method that appears more memory wasteful, where the identity `ID`(*F*) of a fragment *F* is composed of the (unique) identity of its root together with its level. We also need each node *v* to know the weight *ω*(*F*) of the minimum outgoing edge of each fragment *F* containing *v*. To summarize, the *piece of information* `I`(*F*) required in each node *v* per fragment *F* containing *v* is `ID`(*F*) ∘ *ω*(*F*). Thus, `I`(*F*) can be encoded using *O*(log*n*) bits. Again, note that since a node may participate in ℓ = Θ(log*n*) fragments, the total number of bits used for storing all the `I`(*F*) for all fragments *F* containing *v* would thus be Θ(log2*n*). Had no additional steps been taken, this would have violated the *O*(log*n*) memory constraint.\nLuckily, the total number of bits needed globally for representing the pieces `I`(*F*) of all the fragments *F* is only *O*(*n*log*n*), since there are at most 2*n* fragments, and `I`(*F*) of a fragment *F* is of size *O*(log*n*) bits. The difficulty results from the fact that multiple nodes need replicas of the same information. (E.g., all the nodes in a fragment need the `ID` of the fragment.) If a node does not store this information itself, it is not clear how to bring all the many pieces of information to each node who needs them, in a short time (in spite of congestion) and while having only a constant number of such pieces at a node at each given point in time.\nTo allow some node *v* to check whether its neighbour *u* belongs to *v*’s level *j* fragment *F**j*(*v*) for some level *j*, the verifier at *v* needs first to reconstruct the piece of information `I`(*F**j*(*v*)). Intuitively, we had to distribute the information, so that `I`(*F*) is placed “not too far” from every node in *F*. To compare `I`(*F**j*(*v*)) with a neighbour *u*, node *v* must also obtain `I`(*F**j*(*u*)) from *u*. This requires some mechanism to synchronize the reconstructions in neighbouring nodes. Furthermore, the verifier must be able to overcome difficulties resulting from faults, which can corrupt the information stored, as well as the reconstruction and the synchronization mechanisms.\nThe above distribution of the `I`’s is described in Section [sec:proof]. The distributed algorithm for the “fragment by fragment” reconstruction (and synchronization) is described in Section [sub:3.2]. The required verifications for validating the `I`’s and comparing the information of neighbouring nodes are described in Section [sub:utilizing].\n### Distribution of the pieces of information (described in detail in Section [sec:proof])\nAt a very high level description, each node *v* stores permanently `I`(*F*) for a constant number of fragments *F*. Using that, `I`(*F*) is “rotated” so that each node in *F* “sees” `I`(*F*) in *O*(log*n*) time. We term the mechanism that performs this rotation a *train*. A first idea would have been to have a separate train for each fragment *F* that would “carry” the piece `I`(*F*) and would allow all nodes in *F* to see it. However, we did not manage to do that efficiently in terms of time and of space. That is, one train passing a node could delay the other trains that “wish” to pass it. Since neighbouring nodes may share only a subset of their fragments, it is not clear how to pipeline the trains. Hence, those delays could accumulate. Moreover, as detailed later, each train utilizes some (often more than constant) memory per node. Hence, a train per fragment would have prevented us from obtaining an *O*(log*n*) memory solution.\nA more refined idea would have been to partition the tree into connected parts, such that each part *P* intersects *O*(∣*P*∣) fragments. Using such a partition, we could have allocated the *O*(∣*P*∣) pieces (of these *O*(∣*P*∣) fragments), so that each node of *P* would have been assigned only a constant number of such pieces, costing *O*(log*n*) bits per node. Moreover, just one train per part *P* could have sufficed to rotate those pieces among the nodes of *P*. Each node in *P* would have seen all the pieces `I`(*F*) for fragments *F* containing it in *O*(∣*P*∣) time. Hence, this would have been time efficient too, had *P* been small.\nUnfortunately, we did not manage to construct the above partition. However, we managed to obtain a similar construction: we construct *two* partitions of *T*, called `Top` and `Bottom`. We also partitioned the fragments into two kinds: top and bottom fragments. Now, each part *P* of partition `Top` intersects with *O*(∣*P*∣) top fragments (plus any number of bottom fragments). Each part *P* of partition `Bottom` intersects with *O*(∣*P*∣) bottom fragments (plus top fragments that we do not count here). For each part in `Top` (respectively `Bottom`), we shall distribute the information regarding the *O*(∣*P*∣) top (respectively, bottom) fragments it intersects with, so that each node would hold at most a constant number of such pieces of information. Essentially, the pieces of information regarding the corresponding fragments are put in the nodes of the part (permanently) according to a DFS (Depth First Search) order starting at the root of the part. For any node *v*, the two parts containing it encode together the information regarding all fragments containing *v*. Thus, to deliver all relevant information, it suffices to utilize one train per part (and hence, each node participates in two trains only). Furthermore, the partitions are made so that the diameter of each part is *O*(log*n*), which allows each train to quickly pass in all nodes, and hence to deliver the relevant information in short time.\nThe distributed implementation of this distribution of pieces of information, and, in particular, the distributed construction of the two partitions, required us to come up with a new *multi-wave* primitive, enabling an efficient (in *O*(*n*)) time) parallel (i.e., pipelined) executions of Wave&Echo operations on all fragments of Hierarchy $\\cH\\_{\\cM}$.\n### Viewing the pieces of information (described in detail in Section [sub:3.2])\nConsider a node *v* and a fragment *F**j*(*v*) of level *j* containing it. Recall that the information `I`(*F**j*(*v*)) should reside in some node of a part *P* to which *v* belongs. To allow *v* to compare `I`(*F**j*(*v*)) to `I`(*F**j*(*u*)) for a neighbour *u*, both these pieces must somehow be “brought” to *v*. The process handling this task contains several components. The first component is called the *train* which is responsible for moving the pieces of information through *P*’s nodes, such that each node does not hold more than *O*(log*n*) bits at a time, and such that in short time, each node in *P* “sees” all pieces, and in some prescribed order. Essentially, a train is composed of two ingredients. The first ingredient called *convergecast* pipelines the pieces of information in a DFS order towards the root of the part (recall, the pieces of information of the corresponding fragments are initially located according to a DFS order). The second ingredient *broadcasts* the pieces from the root of the part to all nodes in the part. Since the number of pieces is *O*(log*n*) and the diameter of the part is *O*(log*n*), the synchronous environment guarantees that each piece of information is delivered to all nodes of a part in *O*(log*n*) time. On the other hand, in the asynchronous environment some delays may occur, and the delivery time becomes *O*(log2*n*). These time bounds are also required to self-stabilize the trains, by known art, see, e.g. .\nUnfortunately, delivering the necessary pieces of information at each node is not enough, since `I`(*F**j*(*v*)) may arrive at *v* at a different time than `I`(*F**j*(*u*)) arrives at *u*. Recall that *u* and its neighbour *v* need to have these pieces simultaneously in order to compare them (to know whether the edge *e* = (*u*, *v*) is outgoing from *F**j*(*v*)).\nFurther complications arise from the fact that the neighbours of a node *v* may belong to different parts, so different trains pass there. Note that *v* may have many neighbours, and we would not want to synchronize so many trains. Moreover, had we delayed the train at *v* for synchronization, the delays would have accumulated, or even would have caused deadlocks. Hence, we do not delay these trains. Instead, *v* repeatedly samples a piece from its train, and synchronizes the comparison of this piece with pieces sampled by its neighbours, while both trains advance without waiting. Perhaps not surprisingly, this synchronization turns out to be easier in synchronous networks, than in asynchronous ones. Our synchronization mechanism guarantees that each node can compare all pieces `I`(*F**j*(*v*)) with `I`(*F**j*(*u*)) for all neighbours *u* and levels *j* in a short time. Specifically, *O*(log2*n*) time in synchronous environments and *O*(Δlog3*n*) time in asynchronous ones.\n### Local verifications (described in detail in Section [sub:utilizing])\nSo far, with respect to verifying the Minimality property, we have not discussed issues of faults that may complicate the verification. Recall, the verification process must detect if the tree is not an MST. Informally, this must hold despite the fact that the train processes, the partitions, and also, the pieces of information carried by the trains may be corrupted by an adversary. For example, the adversary may change or erase some (or even all) of such pieces corresponding to existing fragments. Moreover, even correct pieces that correspond to existing fragments may not arrive at a node in the case that the adversary corrupted the partitions or the train mechanism.\nIn Section [sub:utilizing], we explain how the verifier does overcome such undesirable phenomena, if they occur. Intuitively, what is detected is not necessarily the fact that a train is corrupted (for example). Instead, what is detected is the state that *some* part is incorrect (either the tree is not an MST, or the train is corrupted, or... etc.). Specifically, we show that if an MST is not represented in the network, this is detected in time *O*(log2*n*) for synchronous environments and time *O*(Δlog3*n*) for asynchronous ones. Note that for a verifier, the ability to detect while assuming any initial configuration means that the verifier is self-stabilizing, since the sole purpose of the verifier is to detect.\nVerifying that *some* two partitions exist is easy. However, verifying that the given partitions are as described in Section [sec:partitions], rather than being two arbitrary partitions generated by an adversary seems difficult. Fortunately, this verification turns out to be unnecessary.\nFirst, as mentioned, it is a known art to self-stabilize the train process. After trains stabilize, we verify that the set of pieces stored in a part (and delivered by the train) includes all the (possibly corrupted) pieces of the form `I`(*F**j*(*v*)), for every *v* in the part and for every *j* such that *v* belongs to a level *j* fragment. Essentially, this is done by verifying at the root *r*(*P*) of a part *P*, that (1) the information regarding fragments arrives at it in a cyclic order (the order in which pieces of information are supposed to be stored in correct instances), (2) the levels of pieces arriving at *r*(*P*) comply with the levels of fragments to which *r*(*P*) belongs to, as indicated by *r*(*P*)’s data-structure. Next, we verify that the time in which each node obtains all the pieces it needs is short. This is guaranteed by the correct train operation, as long as the diameter of parts is *O*(log*n*), and the number of pieces stored permanently at the nodes of the part is *O*(log*n*). Verifying these two properties is accomplished using a 1-proof labelling scheme of size *O*(log*n*), similarly to the schemes described in Examples 2 and 3 ($\\SP$ and $\\EDIAM$, mentioned in Section [sub:examples]).\nFinally, if up to this point, no node raised an alarm, then for each node *v*, the (possibly corrupted) pieces of information corresponding to *v*’s fragments reach *v* in the prescribed time bounds. Now, by the train synchronization process, each node can compare its pieces of information with the ones of its neighbours. Hence, using similar arguments as was used in the *O*(log2*n*)-memory bits verification scheme of, nodes can now detect the case that either one of the pieces of information is corrupted or that *T* is not an MST.\nA synchronous MST construction in *O*(log*n*) bits memory size and *O*(*n*) time\n================================================================================\nIn this section, we describe an MST construction algorithm, called `SYNC\\_MST`, that is both linear in its running time and memory optimal, that is, it runs in *O*(*n*) time and has *O*(log*n*) memory size. We note that this algorithm is not self-stabilizing and its correct operation assumes a synchronous environment. The algorithm will be useful later for two purposes. The first is for distributively assigning the labels of the MST proof labelling scheme, as described in the next section. The second purpose, is to be used as a module in the self-stabilizing MST construction algorithm.\nAs mentioned, the algorithm of Gallager, Humblet, and Spira (GHS) constructs an MST in *O*(*n*log*n*) time. This has been improved by Awerbuch to linear time, using a somewhat involved algorithm. Both algorithms are also efficient in terms of the number of messages they send. The MST construction algorithm described in this section is, basically, a simplification of the GHS algorithm. There are two reasons for why we can simplify that algorithm, and even get a better time complexity. The first reason is that our algorithm is synchronous, whereas GHS (as well as the algorithm by Awerbuch) is designed for asynchronous environments. Our second aid is the fact that we do not care about saving messages (anyhow, we use a shared memory model), while the above mentioned algorithms strive to have an optimal message complexity. Before describing our MST construction algorithm, we recall the main features of the GHS algorithm.\nRecalling the MST algorithm of Gallager, Humblet, and Spira (GHS)\n-----------------------------------------------------------------\nFor full details of GHS, please refer to. GHS uses connected subgraphs of the final MST, called *fragments*. Each node in a fragment, except for the fragment’s root, has a pointer to its parent in the fragment. When the algorithm starts, every node is the root of the singleton fragment including only itself. Each fragment is associated with its *level* (zero for a singleton fragment) and the identity of its root (this is a slight difference from the description in, where a fragment is said to be rooted at an edge). Each fragment *F* searches for its *minimum outgoing edge* *e*min(*F*) = (*v*, *u*). Using the selected edges, fragments are merged to produce larger fragments of larger levels. That is, two or more fragments of some level *j*, possibly together with some fragments of levels lower than *j*, are merged to create a fragment of level *j* + 1. Eventually, there remains only one fragment spanning the graph which is an MST.\nIn more details, each fragment sends an offer (over *e*min(*F*)) to merge with the other fragment *F*ʹ, to which the other endpoint *u* belongs. If *F*ʹ is of a higher level, then *F* is connected to *F*ʹ. That is, the edges in *F* are reoriented so that *F* is now rooted in the endpoint *v* of *e*min(*F*), which becomes a child of the other endpoint *u*.\nIf the level of *F*ʹ is lower, then *F* waits until the level of *F*ʹ grows (see below, the description of “test” messages). The interesting case is when *F* and *F*ʹ are of the same level $\\level$. If *e*min(*F*) = *e*min(*F*ʹ), then *F* and *F*ʹ merge to become one fragment, rooted at, say, the highest `ID` node between *u* and *v*. The level of the merged fragment is set to $\\level+1$.\nThe remaining case, that (w.l.o.g.) *w*(*e*min(*F*)) > *w*(*e*min(*F*ʹ)) does not need a special treatment. When *F* sends *F*ʹ an offer to merge, *F*ʹ may have sent such an offer to some *F*ʺ over *w*(*e*min(*F*ʹ)). Similarly, *F*ʺ may have sent an offer to some *F*‴ (over *w*(*e*min(*F*ʺ))), etc. No cycle can be created in this chain of offers (because of the chain of decreasing weights *w*(*e*min(*F*)) > *w*(*e*min(*F*ʹ)) > *w*(*e*min(*F*ʺ))...). Hence, unless the chain ends with some fragment of a higher level (recall that treating the case that a fragment’s minimum edge leads to a higher level fragment was already discussed), some two fragments in the above chain merge, increasing their level by one. This case (for the fragments of the chain, excluding the two merging fragments) now reduces to the case (discussed previously) that a fragment *F* makes an offer to a fragment of a higher level.\nThe above describes the behavior of fragments. To implement it by nodes, recall that every fragment always has a root. The root conducts Wave&Echo over the fragment to ask nodes to find their own *candidate* edges for the minimum outgoing edge. On receiving the wave (called “find”), each node *v* selects its minimum edge (*v*, *u*) that does not belong yet to the fragment, and has not been “tested” yet (initially, no edge was “tested”). Node *v* sends a “test” message to *u*, to find out whether *u* belongs to *v*’s fragment. The “test” includes the `ID` of *v*’s fragment’s root *r* and its level $\\level$. If the level of *u*’s fragment is at least $\\level$ then *u* answers. In particular, if *u*’s level is $\\level$ and *u*’s fragment root is *r* then *u* sends a “reject” to *v*, causing *v* to conclude that (*v*, *u*) is not outgoing and cannot be a candidate. (Node *u* does not answer, until its level reaches $\\level$, thus, possibly, causing *v*’s fragment to wait.) In the converging wave (called “found”) of the above “find” broadcast, each node *v* passes to its parent only the candidate edge with the minimum weight (among its own candidate and the candidates it received from its children). Node *v* also remembers a pointer to whoever sent it the above candidate. These pointers form a route from *F*ʹ*s* root to the endpoint of *e*min(*F*). The root then sends a message “change-root”, instructing all the nodes on this route (including itself) to reverse their parent pointers. Hence, *F* becomes rooted at the endpoint of *e*min(*F*), who now can send an offer to “connect” over *e*min(*F*).\nAlgorithm `SYNC\\_MST`: a synchronous linear time version with optimal memory size\n---------------------------------------------------------------------------------\nThe algorithm we now describe is synchronous and assumes that all the nodes wake up simultaneously at round 0. However, to keep it easy for readers who are familiar with GHS, we tried to keep it as similar to GHS as possible.\nInitially, each node is a root of a fragment of *level* 0 that contains only itself. During the execution of `SYNC\\_MST`, a node who is not a root, keeps a pointer to its parent. The collection of these pointers (together with all the nodes) defines a forest at all times. Each node also keeps an *estimate* of the `ID` and the level of the root of its fragment. As we shall see later, the `ID` estimate is not always accurate. The level estimate is a lower bound on the actual level. We use the levels for convenience of comparing the algorithm to that of GHS (and for the convenience of the proof). The levels actually can be computed from the round number, or from the counting procedure defined below. More specifically, the algorithm is performed in synchronous *phases*. Phase *i* starts at round 11 ⋅ 2*i*. Each root *r*(*F*) of a fragment *F* (that is, a node whose parent pointer is null) starts the phase by setting its level to *i* and then counting the number of nodes in its fragment.\nThe counting process, called `Count\\_Size`, is defined later, but for now it suffices to say that it consumes *precisely* 2*i* + 2 − 1 rounds. If the diameter of the fragment is small, then some waiting time is added to keep the precise timing. On the other hand, if the number of nodes in the fragment is too large, `Count\\_Size` may terminate before all the nodes in the fragment are counted. Specifically, we guarantee that if the counting process succeeds to count all nodes in the fragment *F* then the precise number of these nodes is known to the root *r*(*F*) at the end of the counting procedure. On the other hand, if the counting process does not count all nodes, then the number of nodes in the fragment is at least 2*i* + 1, and at the end of the `Count\\_Size` process, the root *r*(*F*) learns this fact. Moreover, in such a case, as a consequence, *r*(*F*) changes its level to *i* + 1.\n[def:fragments] A root *r*(*F*) is *active* in phase *i* if and only if ∣*F*∣ ≤ 2*i* + 1 − 1, where ∣*F*∣ denotes the number of nodes in *F*. Note that if *r*(*F*) is active then its level is *i*. In particular, all the roots are *active* in phase (and level) 0. A fragment is *active* when its root is active.\n[com:fragments] When constructing the marker algorithm in later sections, we use the fragments constructed by algorithm `SYNC\\_MST`. More specifically, we refer only to the active fragments. As is easy to observe below in the current section, an active fragment is a specific set of nodes that does not change. This is because when the fragment merges with others (or when others join it), it is no longer the same fragment. In particular, when the new set of nodes will be active, it will be in a higher phase.\n#### Procedure `Find\\_Min\\_Out\\_Edge`:\nConsider the root *r*(*F*) of fragment *F*, who is active in phase *i*. At round (11 + 4) ⋅ 2*i*, each such root *r*(*F*) instructs the nodes in *F* to search for the minimum outgoing edge of *F*. This procedure, called `Find\\_Min\\_Out\\_Edge`, could have been combined with the counting, but we describe it as a separate stage for the sake of simplicity. The method is the same as that of GHS algorithm, except that we achieve an exact timing obtained by not saving in messages. The search is performed over exactly the same set of nodes which has just been counted. This is implemented by a Wave operation initiated by *r*(*F*), carrying *r*(*F*)’s `ID` and level. At precisely round (11 + 6) ⋅ 2*i*, each node *v* who has received the wave, finds the minimum outgoing edge emanating from it. That is, *v* looks at each of its neighbours *u* to see whether *u* belongs to a different fragment of some other root *r*(*F*ʹ) ≠ *r*(*F*). We now describe how *v* identifies this.\nLet us note here two differences from GHS. First, node *v* tests all of its emanating edges at the same time, rather than testing them one by one (as is done in GHS). Moreover, it does not reject any edge, and will test all its edges in the next searches too. Intuitively, the above mentioned one by one process was used in GHS in order to save messages. We do not try to save messages, and the simultaneous testing allows us to keep an exact timing on which we rely heavily. Second, in GHS, node *u*’s estimate of its level may be lower than that of node *v*. In GHS, *v* then needs to wait for *u* to reach *v*’s level, before *v* knows whether edge (*v*, *u*) is outgoing. The main reason this action is useful in GHS is to save on message complexity. Here, again, we do not intend to save messages.\nRecall that the root of *v*’s fragment *F* is active at phase *i*, hence, ∣*F*∣ < 2*i* + 1. (We shall show that no additional nodes joined *F* after the counting.) Hence, at round (11 + 6) ⋅ 2*i*, *all the nodes* in *F* have already received the wave, and set their `ID` estimates to `ID`(*r*(*F*)). The big gain from that, is that at round (11 + 6) ⋅ 2*i*, the `ID`s of the roots of *u* and *v* are different if and only if the edge (*v*, *u*) is outgoing at *v*. The minimum outgoing edge in the fragment of *r*(*F*) is then computed during the convergecast, using the standard Wave&Echo technique. Thus, Procedure `Find\\_Min\\_Out\\_Edge` (composed of the aforementioned Wave&Echo) lasts at most 2(2*i* + 1 − 1) round units, hence (having been started at round (11 + 4) ⋅ 2*i*), it is completed by round (11 + 8) ⋅ 2*i* − 1.\n#### Merging and reorienting edges:\nLet (*w*, *x*) be the chosen minimum outgoing edge from the fragment *F*, such that *w* ∈ *F*. Later, we refer to it as the *candidate* edge. At round (11 + 8) ⋅ 2*i*, an active root *r*(*F*) of *F* starts the process of re-orienting the edges in *F* towards *w*. (For a more thorough description of the root transfer refer to.) This takes at most 2(2*i* − 1) rounds.\nNode *w* then conducts a handshake with *x*, referred to as the *pivot* of *F*. This takes a constant time, but, to keep the total numbers simple, we pad this time to 2*i*. One case is that *w* is, at that time exactly, a pivot of the fragment of *x*, and also `ID`(*x*) < `ID`(*w*). In this case, node *x* will become the child of *w*. In every other case, *w* hooks upon the other endpoint *x* (sets its parent pointer to point at *x*). The hooking is performed exactly at round (11 + 11) ⋅ 2*i* − 1, ending phase *i*. Since the next phase starts at round 22 ⋅ 2*i* there is no overlap between phases.\n#### Procedure `Count\\_Size`:\nTo complete the description of a phase, it is left to describe the counting process, namely, Procedure `Count\\_Size`. To count, a root starts a Wave&Echo, attaching a *time-to-live* = 2*i* + 1 − 1 counter to its broadcast message. A child *c* of a node *y* accepts the wave only if the time-to-live is above zero. Child *c* then copies the wave broadcast message, decrementing the time-to-live (by 1). If, after decrementing, the value of time-to-live is zero, then *c* is a leaf who needs to start the echo. The number of the nodes (who copied the broadcast message) is now counted during the echo in the standard way. Finally, if the count covers the whole graph, this can be easily detected at the time of the echo. The algorithm then terminates.\nTo sum up, phase *i* of the MST construction algorithm is composed of the following components.\n#### \nStarts at round 11 ⋅ 2*i* ;\nRoot *r*(*F*) of each fragment *F* initiates Procedure `Count\\_Size`.\nAt the end of the procedure, we have: \n∣*F*∣ ≤ 2*i* + 1 − 1 iff (1) *r*(*F*) is *active* and (2) all nodes in *F* have their `ID` estimates set to `ID`(*r*(*F*)) ;\nAt round (11 + 4) ⋅ 2*i*, each active root *r*(*F*) initiates Procedure `Find\\_Min\\_Out\\_Edge` ;\nAt round (11 + 8) ⋅ 2*i*, merge fragments and re-orient edges in the newly created fragments.\nThe proof that the collection of parent pointers forms a forest (or a tree) at all times is the same as in GHS. Let us now analyze the round complexity. Observe that each phase *i* takes *O*(2*i*) time. Hence, the linear time complexity of the algorithm follows from the lemma below.\n[lem:disappear] The size of a fragment *F* in phase *i* (and in level *i*) is at least 2*i*. Moreover, ∣*F*∣ < 2*i* + 1 if and only if *r*(*F*) is active by round (11 + 4) ⋅ 2*i*.\nLet us first prove the second part of the lemma. Before deciding whether to be active, a root *r*(*F*) of level *i* counts the number of nodes in its fragment, by employing Procedure `Count\\_Size`. If the count amounts to 2*i* + 1 or more, then the level of *r*(*F*) is set to *i* + 1. Otherwise, we have ∣*F*∣ ≤ 2*i* + 1 − 1 and the root of *F* becomes active. Since Procedure `Count\\_Size` is terminated by round (11 + 4) ⋅ 2*i*, the second part of the lemma follows. To prove the first part of the lemma, we need to show that the size of a level *i* fragment is at least 2*i*. We prove this by induction on *i*.\nIntuitively, the induction says that each fragment at the beginning of phase *i* − 1, is of size at least 2*i* − 1. During phase *i* − 1, by the second part of the lemma, at time (11 + 4) ⋅ 2*i* − 1, all the non-active fragments are already of size at least 2*i* and are also of level *i* (as a result of the count). As for active fragments, each such fragment is combined to at least one other fragment, so the resulting size is at least 2 ⋅ 2*i* − 1 = 2*i*.\nIn more details, note that the claim holds for phase *i* = 0. For a larger phase *i*, assume that the lemma holds for phases up to *i* − 1 including. Consider a root *r*(*F*) of a fragment *F* of level *i*. It was a root also at level *i* − 1. First, assume that at phase *i* − 1, some other root *r*(*F*1) hooked upon *r*(*F*)’s tree. To do so, *r*(*F*1) had to be active at phase *i* − 1. By the induction hypothesis (the first part of the lemma), the size of fragment *F*1, as well as the size of fragment of *F* at that point in time, was at least 2*i* − 1. The claim, in this case, follows.\nNow, assume that no other fragment hooked upon fragment *F* in phase *i* − 1. Note that *F* at level *i* − 1 does not span the graph (otherwise, no root would reach level *i*, by the 2nd part of the inductive hypothesis, and since the counting process on trees is a known art and is known to be correct). Hence, it has a minimum outgoing edge *e* = (*w*, *x*), where *w* ∈ *F* and *x* belongs to some other fragment *F**x*. We claim that the search process `Find\\_Min\\_Out\\_Edge` does find that edge *e*. Recall that in the fragment of an active root, the counting reaches all the nodes in the fragment. Hence, each of them knows the `ID` of their root *r*(*F*) at the time the search in their fragment starts. Moreover, since a hooking is performed only at times of the form (11 + 11) ⋅ 2*j* − 1, no new nodes (or fragments) join until the last time step of the phase (which is after the search, because of what we established about the size of the fragment).\nWe claim that either (a) Edge *e* = (*w*, *x*) was not the minimum outgoing edge of *x*’s fragment *F**x*, or, alternatively, (b) the root *r**x* of *x*’s fragment *F**x* had level *i* at that time, or (c) `ID`(*x*) > `ID`(*w*). Assume the contrary. By the inductive hypothesis, node *r**x* is at least at level *i* − 1. Since we assumed that (b) does not hold, *r**x* is exactly at level *i* − 1. This means (by the correctness of the counting) that *r**x* is active at phase *i* − 1. Similarly, this also means that the size of *F**x* is less than 2*i*. Hence, and by the induction hypothesis, the counting, the searching, the root transfer, and the handshake end in *x*’s fragment at the same time they end in *w*’s fragment. (These processes use known art, and we shall not prove them here.) Then *x* hooks upon *w*, contrary to our assumption.\nWe have just established that the conditions for *w* to hook upon *x* hold. Hence, *w* hooks upon *x*. Similarly to the previous case, the size of *w*’s fragment, as well as the size of *x*’s fragments at that point in time is at least 2*i* − 1. Thus, the size of the combined fragment is at least 2*i*. This concludes the proof of the first part of the lemma.\n[cor:time-linear] The synchronous MST construction algorithm computes an MST in time *O*(*n*).\n#### Implementing the algorithm in the shared memory model with *O*(log*n*) memory:\nEach node *v* keeps its fragment level and root `ID`. Node *v* also remembers whether *v* is in the stage of counting the number of nodes, or searching for the minimum outgoing edge. It also needs to remember whether it is in the wave stage, or has already sent the echo. Node *v* needs to remember the candidate (for being *e*min(*F*)) edge that *v* computed in the echo (“found”) stage of the convergecast. If this candidate was reported by a child, then *v* also remembers a pointer to that child. Clearly, all the above variables combined need *O*(log*n*) bits of memory.\nAt a first glimpse it may look as if a node must also remember the list of pointers to its children. The list is used for (1) sending the wave (e.g., the “find” message of the search, using GHS terms) to all the children, and (2) knowing when all the children answered the echo (e.g., the “found” of the search). Note that a node does not need to store this list itself. Node *v* can look at each neighbour *u* to see whether the neighbour is *v*’s child. (For that purpose, if *u* is a child of *v*, then *u* stores *v*’s `ID` as *u*’s `Parent\\_ID`.) Clearly, this can be implemented using *O*(log*n*) bits per node.\nTo implement (1), node *v* posts its wave broadcast message (e.g., the “find”) so that every neighbour can read it. However, only its children actually do. To allow the implementation of (2), a precaution is taken before the above posting. Node *v* first posts a request for its children to reset their `ECHO` variables, and performs the posting of “find” only when it sees that `ECHO` has been reset for every neighbour *w* whose parent pointer points at *v*.\nTo implement (2), node *v* further reads its neighbours repeatedly. It knows all its children echoed the wave in an iteration when it has just finished rereading all its neighbours, and every node *u* pointing at *v* (as its parent), also sets its `ECHO` variable to some candidate edge (or to some default value if it has no candidate edge).\n[obs:spce-sync] The space required by the linear time synchronous algorithm is *O*(log*n*) bits per node.\nThe theorem below follows from Observation [obs:spce-sync] and Corollary [cor:time-linear].\n[thm:ALG] The synchronous algorithm `SYNC\\_MST` computes an MST in time *O*(*n*) and memory size *O*(log*n*).\nRepresenting and verifying a hierarchy\n======================================\nWe are now ready to describe our proof labeling scheme $(\\cM,\\cD)$ for MST. The goal of this section is to construct some part of the marker $\\cM$, and the corresponding part of the verifier $\\cD$, which are relatively easy to construct. The techniques used in this section bear similarity to the techniques presented in. Hence, we only expose the main ideas behind this part of the proof labeling scheme, leaving out some of the technicalities. Nevertheless, since the notion of proof labeling schemes can sometimes be confusing, this section may help the reader to get accustomed to the notion and the difficulties that may arise.\nAs a warm up, we first note that using the 1-proof labeling scheme described in Example $\\SP$, we may assume that *H*(*G*) ≡ *T* is a spanning tree of *G* rooted at some node *r*, and that each node knows which of its neighbours in *G* are its children in *T* and which is its parent. Moreover, using the 1-proof labeling scheme described in Example $\\NK$, we may also assume that each node knows *n*. The 1-proof labeling schemes described in Examples $\\SP$ and $\\NK$ use *O*(log*n*) memory size and can be constructed using *O*(*n*) time. Hence, using them does not violate the desired complexity constrains of our scheme. Thus, from now on, let us fix a spanning tree *T* = (*V*(*G*), *E*(*T*)) of a graph *G* = (*V*(*G*), *E*(*G*)), rooted at some node *r*(*T*). The goal of the rest of the verification scheme is to verify that *T* is in fact, minimal. Before we continue, we need a few definitions.\n[def:hierarchy] A *hierarchy* $\\cH$ for *T* is a collection of fragments of *T* satisfying the following two properties.\n1. $T\\in\\cH$ and, for every *v* ∈ *V*(*G*), there is an $F\\_v\\in\\cH$ such such *V*(*F**v*) = {*v*} and *E*(*F**v*) = ∅.\n2. For any two fragments *F* and *F*ʹ in $\\cH$, if *F* ∩ *F*ʹ ≠ ∅ then either *F* ⊆ *F*ʹ or *F*ʹ ⊆ *F*. (That is, the collection of fragments is a laminar family.)\nPlease recall (Definition [com:fragments] and Comment [com:fragments]) that when we construct a hierarchy according to Definition [def:hierarchy], the fragments referred to are the active fragments constructed in `SYNC\\_MST`.\nThe *root* of a fragment *F* is the node in *F* closest to the root of *T*. For a fragment $F\\in\\cH$, let $\\cH(F)$ denote the collection of fragments in $\\cH$ which are *strictly* contained in *F*. Observe that a hierarchy $\\cH$ can be viewed as a rooted tree, whose root is the fragment *T*, and whose leaves are the singleton fragments in $\\cH$. A child of a non-singleton fragment $F\\in \\cH$ is a fragment $F'\\in\\cH(F)$ such that no other fragment $F''\\in \\cH(F)$ satisfies *F*ʺ ⊃ *F*ʹ. Note that the rooted tree induced by a hierarchy is unique (if the children are unordered). To avoid confusion with tree *T*, we use the name *hierarchy-tree* (or, sometimes even just *hierarchy*) for the above mentioned tree induced by a hierarchy. We associate a *level*, denoted *l**e**v*(*F*), with each fragment $F\\in \\cH$. It is defined as the height of the node corresponding to *F* in the hierarchy-tree induced by $\\cH$, i.e., the maximal number of fragments on a simple path in $\\cH$ connecting *F* to a singleton fragment. In particular, the level of a singleton fragment is 0. The level of the fragment *T* is called the *height* of the hierarchy, and is denoted by ℓ. Figure [fig:HierarchyTree1] depicts a hierarchy $\\cH$ of a tree *T*.\n [fig:HierarchyTree1]\nGiven a hierarchy $\\cH$ for a spanning tree *T*, a function $\\chi:\\cH\\setminus\\{T\\} \\longrightarrow E(T)$ is called a *candidate function* of $\\cal{H}$ if it satisfies $E(F)=\\{ \\chi(F') | F'\\in \\cH(F) \\}$ for every $F\\in\\cH$. (Less formally, *F* is precisely the union of the candidate edges *χ*(*F*ʹ) of all fragments *F*ʹ of $\\cH$ strictly contained in *F*).\nThe proof of the following lemma is similar, e.g., to the proof of.\n[lem:construction] Let *T* be a spanning tree of a graph *G*. If there exists a candidate function *χ* for a hierarchy $\\cH$ for *T*, such that for every $F\\in \\cH$, the candidate edge *χ*(*F*) is a minimum outgoing edge from *F*, then *T* is an MST of *G*.\nWe prove the claim that each fragment $F\\in\\cH$ is a subtree of an MST of *G*, by induction on the level *l**e**v*(*F*) of fragment *F*. Note that the claim obviously holds for any fragment *F* with *l**e**v*(*F*) = 0 since *F* is a singleton fragment.\nNow consider a fragment *F* with *l**e**v*(*F*) = *k* under the inductive assumption that the claim holds for every fragment *F*ʹ with *l**e**v*(*F*ʹ) < *k*. Let *F*1, *F*2, …, *F**a* be the child fragments of *F* in $\\cH$. Since for each *i* ∈ [1, *a*], fragment *F**i* satisfies *l**e**v*(*F**i*) < *k*, the induction hypothesis implies that *F**i* is a subtree of the MST. It also follows from the facts that $E(F)=\\{ \\chi(F') | F'\\in \\cH(F) \\}$ and $E(F\\_i)=\\{ \\chi(F') | F'\\in \\cH(F\\_i) \\}$ for each *i* ∈ [1, *a*] that the fragment *F* is obtained by connecting *F*1, *F*2, …, *F**a* with their minimum outgoing edges. In the case that a fragment *F*ʹ is a fragment of an MST (as is the case here for *F*1, *F*2, …, *F**a*, by the induction hypothesis), it is known that the union of *E*(*F*ʹ) with the minimum outgoing edge of *F*ʹ is a fragment of the MST (the “safe edge” theorem). (See e.g.,.) Thus, fragment *F*, which is obtained by connecting fragments *F*1, *F*2, …, *F**a* with their minimum outgoing edges, is a subgraph of an MST.\nInformally, suppose that we are given distributed structures that are claimed to be a tree *T*, a “legal” hierarchy $\\cH$ for the tree, and a “legal” candidate function for the hierarchy. The goal obtained in the current section is to verify the following properties of these structures. First, verify that this indeed is a hierarchy for *T* of height ℓ ≤ ⌈log*n*⌉ and a candidate function *χ* for $\\cH$. Moreover, verify that each node *v* “knows” to which levels of fragments *v* belongs and which of its neighbours in *T* share the same given fragment. (Note that this section does not guarantee that knowledge for neighbours in *G* who are not neighbours in *T*.) In addition, each node is verified to “know” whether it is adjacent to a candidate edge of any of the fragments it belongs to. Put more formally, this section establishes the following lemma.\n[lem:simple-proof] There exists a 1-proof labeling scheme with memory size *O*(log*n*) and construction time *O*(*n*) that verifies the following:\n*H*(*G*) ≡ *T* is a spanning tree of *G* rooted at some node *r*, and each node knowns *n*.\nThe cartesian product of the data-structures indeed implies a hierarchy $\\cH$ for *T* of height ℓ ≤ ⌈log*n*⌉ and a candidate function *χ* for $\\cH$. Furthermore, the data-structure at each node *v* allows it to know,\nWhether *v* belongs to a fragment *F**j*(*v*) of level *j* in $\\cH$ for each 0 ≤ *j* ≤ ℓ, and if so:\nWhether *v* is the root of *F**j*(*v*).\nWhether *v* is an endpoint of the (unique) candidate edge of *F**j*(*v*), and if so, which of the edges adjacent to *v* is the candidate edge.\nGiven the data-structure of a node *u* which is a neighbour of *v* in *G*, i.e., (*v*, *u*) ∈ *E*(*G*), node *v* can find out whether they are neighbours in *T* as well, i.e., whether (*u*, *v*) ∈ *E*(*T*), and if so, for each 1 ≤ *j* ≤ ℓ, whether *u* belongs to *F**j*(*v*).\nHierarchy $\\cH\\_{\\cM}$ and candidate function $\\chi\\_{\\cM}$\n-----------------------------------------------------------\nOn a correct instance, i.e., when *T* is indeed an MST, the marker $\\cM$ first constructs a particular hierarchy $\\cH\\_{\\cM}$ over *T* and a candidate function $\\chi\\_{\\cM}$ for that hierarchy. Hierarchy $\\cH\\_{\\cM}$ and candidate function $\\chi\\_{\\cM}$ are designed so that indeed each candidate of a fragment is a minimum outgoing edge from that fragment. The marker then encodes hierarchy $\\cH\\_{\\cM}$ and candidate function $\\chi\\_{\\cM}$ in one designated part of the labels using *O*(log*n*) bits per node. Note, however, that these bits of information may be corrupted by the adversary. We will therefore need to employ another procedure that verifies that indeed a hierarchy $\\cH$ and a candidate function *χ* are represented by the cartesian product of the encodings of all nodes. By Lemma [lem:construction], it is not necessary that the verifier checks that $\\cH$ is, in fact, the particular hierarchy $\\cH\\_{\\cM}$ constructed by the marker, or that the candidate function *χ* is $\\chi\\_{\\cM}$. However, as is clear from the same lemma, we do need to show that $\\cH$ and candidate function *χ* satisfy that indeed each candidate of a fragment is a minimum outgoing edge from that fragment. This task is the main technical difficulty of the paper, and is left for the following sections.\nThe hierarchy $\\cH\\_{\\cM}$ and Candidate function $\\chi\\_{\\cM}$ built by the marker algorithm are based on `SYNC\\_MST`, the new MST construction algorithm described in Section [sec:mst-construction]. Since we assume that the MST is unique, Algorithm `SYNC\\_MST` will in fact construct the given MST. (Recall that we describe here the labels assigned by the marker to a correct instance, where the given subgraph *T* is indeed an MST.) The hierarchy and candidate function we define for *T* follow the merging of *active* fragments in algorithm `SYNC\\_MST`. More precisely, the nodes in $\\cH\\_{\\cM}$ are the active fragments defined during the execution of `SYNC\\_MST`. Recall from Section [sec:mst-construction], that an active fragment *F* joins some fragment *H* of *T*, through its minimal outgoing edge *e*. (It is possible that at the time *F* joins *H*, *H* itself was an active fragment that joined *F* through its own minimal outgoing edge that is also *e*.) Note that with time, some other fragments join the resulted connected component, until, at some point, the resulted connected component becomes an active fragment *F*ʹ. In the hierarchy tree $\\cH\\_{\\cM}$, fragment *F* is defined as the child of *F*ʹ, and the candidate edge of *F* is *e*, i.e., *χ*(*F*) = *e*.\nAs proved in Lemma [lem:disappear], after performing the algorithm for level *i*, the size of every fragment is at least 2*i*. Thus, in particular, the height of the hierarchy $\\cH$ is at most ⌈log*n*⌉. The candidate function $\\chi\\_{\\cM}$ chosen by the marker for $\\cH\\_{\\cM}$ is defined by the minimum outgoing edges selected by the algorithm, i.e., for each $F\\in\\cH\\_{\\cM}$, the candidate edge *χ*(*F*) is the selected edge of *F*. Thus, under $\\chi\\_{\\cM}$, each candidate of a fragment is, actually, a minimum outgoing edge.\nRepresenting a hierarchy distributively and verifying it locally\n----------------------------------------------------------------\n[app:hierarchy]\n#### Representing a hierarchy:\nLet ℓ ≤ ⌈log*n*⌉. Given a hierarchy of fragments $\\cH$ of height ℓ over the rooted tree *T* = *H*(*G*), we now describe how we represent it in a distributed manner. Each node *v* keeps a string named `Roots`(*v*) of length ℓ + 1, where each entry in that string is either “1”, “0”, or “\\*”. To be consistent with the levels, we enumerate the entries of each string from left to right, starting at position 0, and ending at position ℓ. Fix *j* ∈ [0, ℓ]. Informally, the *i*’th entry of `Roots`(*v*), namely, `Roots`*i*(*v*), is interpreted as follows.\n`Roots`*i*(*v*) = 1 indicates that *v* is the root of the level *i* fragment it belongs to.\n`Roots`*i*(*v*) = 0 indicates that *v* is not the root of the level *i* fragment it belongs to.\n`Roots`*i*(*v*) = \\* indicates that there is no level *i* fragment that *v* belongs to.\nSee Table [tab:table1] for an example of `Roots` strings of nodes corresponding to Figure [fig:HierarchyTree1].\n|c|ccccc|p1cm|c|ccccc| `Roots` & 0 & 1 & 2 & 3 & 4 & & `EndP` & 0 & 1 & 2 & 3 & 4 \n a & 1 & 0 & 0 & 0 & 0 & & a & up & none & none & none & none \nb & 1 & 1 & 0 & 0 & 0 & & b & down & up & none & none & none \nc & 1 & 0 & 0 & 0 & 0 & & c & up & none & none & none & none \nd & 1 & \\* & 0 & 0 & 0 & & d & up & \\* & none & none & none \ne & 1 & \\* & 0 & 0 & 0 & & e & up & \\* & none & none & none \nf & 1 & 0 & 0 & 0 & 0 & & f & up & down & none & none & none \ng & 1 & 1 & 1 & 1 & 0 & & g & down & none & down & up & none \nh & 1 & \\* & 1 & 0 & 0 & & h & down & \\* & up & none & none \ni & 1 & \\* & 0 & 0 & 0 & & i & up & \\* & none & none & none \nj & 1 & 0 & 0 & 0 & 0 & & j & up & none & none & none & none \nk & 1 & 1 & 1 & 0 & 0 & & k & down & down & up & none & none \nl & 1 & 1 & 1 & 1 & 1 & & l & down & down & down & down & none \nm & 1 & 1 & 0 & 0 & 0 & & m & down & up & none & none & none \nn & 1 & 0 & 0 & 0 & 0 & & n & up & none & none & none & none \no & 1 & 0 & 0 & 0 & 0 & & o & up & none & none & none & none \np & 1 & 1 & 0 & 0 & 0 & & p & down & up & none & none & none \nq & 1 & 0 & 0 & 0 & 0 & & q & up & none & none & none & none \nr & 1 & 0 & 0 & 0 & 0 & & r & up & none & none & none & none \n|c|ccccc|p1cm|c|ccccc| `Parents` & 0 & 1 & 2 & 3 & 4 & & `Or-EndP` & 0 & 1 & 2 & 3 & 4 \n a & 1 & 0 & 0 & 0 & 0 & & a & 1 & 0 & 0 & 0 & 0 \nb & 0 & 1 & 0 & 0 & 0 & & b & 1 & 1 & 0 & 0 & 0 \nc & 0 & 0 & 0 & 0 & 0 & & c & 1 & 0 & 0 & 0 & 0 \nd & 1 & 0 & 0 & 0 & 0 & & d & 1 & 0 & 0 & 0 & 0 \ne & 0 & 0 & 0 & 0 & 0 & & e & 1 & 0 & 0 & 0 & 0 \nf & 1 & 0 & 0 & 0 & 0 & & f & 1 & 1 & 0 & 0 & 0 \ng & 0 & 0 & 0 & 1 & 0 & & g & 1 & 1 & 1 & 1 & 0 \nh & 0 & 0 & 1 & 0 & 0 & & h & 1 & 0 & 1 & 0 & 0 \ni & 0 & 0 & 0 & 0 & 0 & & i & 1 & 0 & 0 & 0 & 0 \nj & 1 & 0 & 0 & 0 & 0 & & j & 1 & 0 & 0 & 0 & 0 \nk & 0 & 0 & 1 & 0 & 0 & & k & 1 & 1 & 1 & 0 & 0 \nl & 0 & 0 & 0 & 0 & 0 & & l & 1 & 1 & 1 & 1 & 0 \nm & 0 & 1 & 0 & 0 & 0 & & m & 1 & 1 & 0 & 0 & 0 \nn & 0 & 0 & 0 & 0 & 0 & & n & 1 & 0 & 0 & 0 & 0 \no & 1 & 0 & 0 & 0 & 0 & & o & 1 & 0 & 0 & 0 & 0 \np & 0 & 1 & 0 & 0 & 0 & & p & 1 & 1 & 0 & 0 & 0 \nq & 1 & 0 & 0 & 0 & 0 & & q & 1 & 0 & 0 & 0 & 0 \nr & 1 & 0 & 0 & 0 & 0 & & r & 1 & 0 & 0 & 0 & 0 \n[tab:table1]\n#### Verifying a hierarchy:\nObserve, the `Roots` strings assigned for a correct instance satisfy the following. \n**(RS0)** The prefix of the `Roots` string at every node is in [1,\\*]\\* and its suffix is in [0,\\*]\\*, \n(\\*because each node is a root of a level 0 fragment and continues being a root in its fragment until some level when it stops (if it does stop); when the node stops being a root, it never becomes a root again\\*)\n**(RS1)** the length of each `Roots` string is ℓ + 1, \n(\\*because there cannot be more than ℓ + 1 levels \\*)\n**(RS2)** the `Roots` string of the root *r* of *T* contains only “1”s and “\\*”s, and its ℓ’th entry is “1”, \n(\\*because a zero in the *i*th position would have meant that *r* is not the root of its fragment of level *i*; the second part follows from the fact that the whole tree is a fragment of level ℓ and *r* is its root \\*)\n**(RS3)** the first entry (at position 0) of every `Roots` string is “1”, \n(\\*because every node *v* is the root of a singleton fragment of level 0 containing only node *v* \\*),\n**(RS4)** the ℓ’th entry of every non-root node is “0”, \n(\\*because only *r* is the root of a fragment of level ℓ, since that fragment is the whole tree \\*)\n**(RS5)** if the *j*’th entry of `Roots`(*v*) is “0” for some node *v* and *j* ∈ [0, ℓ], then the *j*’th entry of the `Roots` strings of *v*’s parent in *T* is not “\\*”. \n(\\*because node *v* belongs to a fragment *F* of level *j*, but is not *F*’s root; hence, *v*’s parent belongs to *F* of level *j* too \\*)\nIt is easy to see that for any assignment of `Roots` strings $\\cI$ obeying rules RS1–RS5 there exists a unique hierarchy whose distributed representation is $\\cI$. Hence, we say that an assignment of `Roots` strings to the nodes of *T* is *legal* if the strings obey the five `Roots` strings conditions above, namely RS1–RS5. For a given legal assignment of `Roots` strings $\\cI$, we refer to its induced hierarchy as the *`Roots`-hierarchy* of $\\cI$. Recall that at this point, we may assume that each node *v* knows the value of *n*, and that each node knows whether it is the root of *T*. Hence, verifying that a given assignment of `Roots` strings is a legal one can be done locally, by letting each node look at its own string and the string of its parent only.\n#### Identifying tree-neighbours in the same fragment:\nObviously, for correct instances, the marker produces a legal assignment of `Roots` strings. For a general instance, if the verifier at some node finds that the assignment of `Roots` is not legal then it raises an alarm. Thus, (if no node raises an alarm), we may assume that hierarchy *`Roots`-hierarchy* exists, and that each node knows (by looking at its own label and the labels of its neighbours in the tree *T*), for every level 0 ≤ *j* ≤ ℓ,\n1. whether it belongs to a fragment *F**j* of level *j*, and if so:\n2. which of its neighbours in *T* belongs to *F**j*.\nRepresenting and verifying a candidate function for the `Roots`-hierarchy\n-------------------------------------------------------------------------\nHaving discussed the proof labeling for the hierarchy, we now describe the proof labeling scheme for the candidate function. Consider now a correct instance *G* and the hierarchy $\\cH\\_{\\cM}$ produced by the marker algorithm. Recall, the candidate function $\\chi\\_{\\cM}$ is given by the selected outgoing edges, which are precisely the minimum outgoing edges of the corresponding fragments, as identified by the construction algorithm `SYNC\\_MST`. We would like to represent this candidate function $\\chi\\_{\\cM}$ distributively, and to verify that this representation indeed forms a candidate function. Moreover, we would make sure that each node *v* be able to know, for each fragment *F* containing it, whether it is an endpoint of the selected edge of *F*, and if so, which of its edges is the selected edge.\n#### Representing a candidate function:\nGiven a correct instance, and its corresponding legal assignment of `Roots` strings, we augment it by adding, for each node *v*, an additional string of ℓ + 1 entries named `EndP`(*v*). Intuitively, `EndP`(*v*) is used by the marker algorithm to mark the levels of the fragments for which *v* is the endpoint of the minimum outgoing edge. Moreover, in a sense, `EndP`(*v*) also is a part of the marking of the specific edge of *v* that is the minimum outgoing edge in that level (in the case that *v* is indeed the endpoint). Let us now give the specific definition of that marking.\nEach entry in `EndP`(*v*) is one of four symbols, namely, “up”, “down”, “none” and “\\*”. The entries of `EndP`(*v*) are defined as follows. Fix an integer *j* ∈ [0, ℓ] and a node *v*. If *v* does not belong to a fragment of level *j* in $\\cH\\_{\\cM}$, then the *j*’th entry in `EndP`(*v*) is “\\*”. Consider now *j* ∈ [0, ℓ] such that *v* does belong to a fragment $F\\in \\cH\\_{\\cM}$ of level *j*. If *v* is not an endpoint of the candidate $\\chi\\_{\\cM}(F)$ of *F*, then the *j*’th entry of `EndP`(*v*) is “none”. Otherwise, node *v* is an endpoint of $\\chi\\_{\\cM}(F)$, i.e., $\\chi\\_{\\cM}(F)=(v,u)$ (for some *u* that is not in *F*). Consider two cases. If *u* is *v*’s parent in *T* then the *j*’th entry of `EndP`(*v*) is set to “up”. If, on the other hand, *u* is a child of *v* in *T*, then the *j*’th entry of `EndP`(*v*) is set to “down”. See Table 1 for an example of `EndP` strings of nodes corresponding to Figure [fig:HierarchyTree1].\nConsider now a node *v* that belongs to a level *j* fragment $F\\in \\cH\\_{\\cM}$. By inspecting its own label, node *v* can find out whether it is an endpoint of a candidate of *F* (recall, from the previous subsection, that it also knows whether or not it belongs to a level *j* fragment). Moreover, in this case, we would like *v* to actually be able to identify in one time unit, which of its incident (tree) edges is the candidate. Obviously, if the *j*’th entry in `EndP`(*v*) is “up”, then the candidate *e* is the edge leading from *v* to its parent in *T*. Intuitively, in the case that the entry is “down”, we would like to store this information in *v*’s children to save space in *v* (since *v* may be the endpoint of minimum outgoing edges for several fragments, of several levels, and may not have enough space to represent all of them). Hence, we attach to each node *x* another string called `Parents`(*x*), composed also of ℓ + 1 bits. For *j* ∈ [0, ℓ], the *j*’th bit in `Parents`(*x*) is “1” if and only if (*y*, *x*) is the candidate of the level *j* fragment that contains *y* (if one exists), where *y* is the parent of *x*. See Table 1 for an example of `Parents` strings of nodes corresponding to Figure [fig:HierarchyTree1]. Now, to identify *u*, node *v* needs only to inspect the `Parents` strings of its children. In either of the above cases for the `EndP`(*v*) entry (“up” or “down”), we call *e* the *induced candidate* of *F*.\n#### Verifying a candidate function:\nGiven a legal assignment of `Roots` strings, we say that assignments of `EndP` and `Parents` strings are *legal* if the following conditions hold:\n**(EPS0)** If the *j*’th entry of `Parents`(*v*) is “1” and *u* is the parent of *v*, then the *j*’th entry of `EndP`(*u*) is “down”, \n(\\* because if *v* indicates the minimum outgoing edge of *u*’s fragment (of level *j*) leads from *u* to *v*, then *v*’s parent *u* indicates this edge leads to one of *u*’s children \\*)\n**(EPS1)** for each fragment *F* of level 0 ≤ *j* < ℓ in the `Roots`-hierarchy, there exists precisely one node *v* ∈ *F* whose *j*’th entry in `EndP`(*v*) is either “up” or “down”, \n(\\*because only one node *v* in each fragment *F* of level *j* is the endpoint of the outgoing edge of *F* \\*)\n**(EPS2)** for each node *v*, if the *j*’th entry in `EndP`(*v*) string is “down” then there exists precisely one child *u* of *v* such that the *j*’th entry in `Parents`(*u*) is “1”, \n(\\*because the j’th entry in `EndP`(*v*) being “down” indicates its minimum outgoing edge leads to *one* of *v*’s children (only *one*, since there is only one minimum outgoing edge of the fragment *F* of level *j* containing *v*); to remember which child, we mark this child *u* by `Parents`(*u*) = 1 \\*)\n**(EPS3)** for each node *v*, and for each 0 ≤ *j* < ℓ, if the *j*’th entry in `EndP`(*v*) string is “up” then:\n1. the *j*’th entry of *v*’s `Roots`-string is “1”, \n(\\*because node *v* belongs to a different fragment *F**v* of level *j* than the level *j* fragment of *v*’s parent; hence, *v* is the highest (closest to the root of the whole tree) in *F**v*, that is, *v* is *F**v*’s root \\*)\n2. [item-contain] for every *i* > *j*, the *i*’th entry of *v*’s `Roots`-string is not “1”, \n(\\*because fragment *F**v* of *v* in level *j* merges with the fragment (of level *j*) of *v*’s parent; hence, *v* is not the highest in its fragments of levels *i* > *j*\\*)\n**(EPS4)** if the *j*’th entry in `Parents`(*v*) is “1” then:\n1. the *j*’th entry of *v*’s `Roots`-string is not “0”, \n(\\*because node *v* is not in the fragments of level *j* of *v*’s parent (see EPS2); hence, either *v* is the root of its fragment of level *j* (see EPS3, part 1), or *v* does not belong to a fragment of level *j* \\*)\n2. for every *i* > *j*, the *i*’th entry of *v*’s `Roots`-string is not “1”, \n(\\*See EPS3 part 2 \\*)\n[item-every] **(EPS5)** for every non-root node *v*, there exists an index integer *j* ∈ [0, ℓ], such that either the *j*’th entry in `Parents`(*v*) is 1 or the *j*’th entry in `EndP`(*v*) is “up”. \n(\\*because every node is the root of a fragment of level 0; at some level, *v*’s fragment merges with the fragment of *v*’s parent \\*)\n[lem:correct-candidate] Consider a `Roots`-hierarchy $\\cH$ given by a legal assignment of `Roots` strings. The conditions EPS1–EPS5 above imply that legal assignments of `EndP` and `Parents` strings (with respect to $\\cH$) induce a candidate function $\\chi:\\cH\\setminus\\{T\\} \\longrightarrow E(T)$.\nCondition EPS1 implies that for each fragment *F* ≠ *T*, there is precisely one node “suspected” as an endpoint of the induced candidate of *F*. Condition EPS2 together with the previous one, implies that there is precisely one induced candidate edge *χ*(*F*) for each fragment *F* ≠ *T*. That is, these two conditions induce a function $\\chi:\\cH\\setminus\\{T\\} \\longrightarrow E(T)$. Our goal now is to show that *χ* is, in fact, a candidate function. That is, we need to show that for every fragment $F\\in\\cH$, we have $E(F)=\\{ \\chi(F') | F'\\in \\cH(F) \\}$. (Recall, $\\cH(F)$ denotes the set of fragments in $\\cH$ which are strictly contained in *F*.)\nIt follows by the second items in Conditions EPS3 and EPS4, that for every fragment $F\\in\\cH$, we have\n \n$$E(F)\\supseteq \\{ \\chi(F') | F'\\in \\cH(F) \\}$$ \nIn particular, we have $E(T)\\supseteq \\{ \\chi(F') | F'\\in \\cH(T) \\}$. Now, by Condition EPS5, we get that each edge of *T* is an induced candidate of some fragment. That is, we have:\n \n$$E(T)=\\{ \\chi(F') | F' \\in \\cH(T) \\}$$ \nThe first items in Conditions EPS3 and EPS4 imply that for every fragment $F\\in \\cH\\setminus\\{T\\}$, the edge *χ*(*F*) is outgoing from *F*. This fact, together with part (2) in the definition of a hierarchy, implies that for every fragment $F\\in\\cH$,\n \n$$\\{ \\chi(F') | F' \\not \\in \\cH(F) \\} \\bigcap E(F) ~=~\\emptyset.$$ \nEquations (1), (2), and (3) imply that for every fragment $F\\in\\cH$, $\\{ \\chi(F') | F'\\in \\cH(F) \\}$. In other words, *χ* is a candidate function for $\\cH$, as desired.\nCondition EPS0 is not required in order to prove the above lemma. If the labels were assigned by our MST construction algorithm, condition EPS0 holds too. Even though adding the condition seems redundant, we decided to add it because we believe it makes the reading more intuitive.\nNow, to verify that assignments of `EndP` and `Parents` strings are legal with respect to a given legal assignment of `Roots` strings, we need to verify the five conditions above. Conditions EPS2–EPS5 can be verified easily, in 1 unit of time, while the first condition EPS1 needs additional information at each node to be verified in 1 unit of time. Specifically, verifying the rule amounts to verifying that exactly one of the nodes in a fragment of level *i* has its *i*’th position in `EndP` equal to 1. This is easy to do in a scheme that is very similar to Example $\\NK$ in Section [sub:examples]. Hence, we omit this simple description (nevertheless, it is demonstrated in Table [tab:table1] in the example of the `Or\\_EndP` strings of nodes corresponding to Figure [fig:HierarchyTree1]).\nThe distributed marker algorithm\n--------------------------------\n[app:marker] A natural method for assigning the labels of the 1-proof labeling scheme described above (composed of the representation of $\\cH\\_{\\cM}$ and $\\chi\\_{\\cM}$, and the strings `Roots`, `Parents`, `EndP`, and `Or\\_EndP`), is to follow the construction algorithm of the MST, namely `SYNC\\_MST` (see Section [sec:mst-construction]), which, in particular, constructs the hierarchy $\\cH\\_{\\cM}$ and the candidate function $\\chi\\_{\\cM}$. Recall that the complexity of `SYNC\\_MST` is *O*(*n*) time and *O*(log*n*) bits of memory per node.\nEssentially, assigning the labels is performed by adding some set of actions to `SYNC\\_MST`. These actions do not change the values of any of the variables of the original algorithm. Also, we do not change the algorithm’s flow of control, except for adding these actions. Since each action is just a new assignment to a new variable (of logarithmic size), the addition of these actions cannot violate the correctness of `SYNC\\_MST`, nor change its time and memory complexities (except by a constant factor). We note that adding these actions on top of `SYNC\\_MST` is not complicated, and can be realized using standard techniques. Hence, we omit it here. Hence, we obtain the following.\n[lem:lineal-marker] There exists a distributed marker algorithm assigning the labels of the 1-proof labeling scheme described in Section [sec:section5], running in *O*(*n*) time and using *O*(log*n*) bits of memory per node.\nThe lemma above together with Lemma [lem:correct-candidate] establishes Lemma [lem:simple-proof].\nDistributing pieces of information\n==================================\nIn the previous section, we described the verification that (1) a tree exists, (2) it is decomposed into a hierarchy of fragments, and (3) edges emanating from the fragments compose a candidate function (so that the tree is the collection of these edges). That verified the Well-Forming property. It is left to verify the Minimality property. That is, it is left to show that each edge of the candidate function is the minimum outgoing edge of some fragment in the hierarchy. The current section describes a part of the marker algorithm responsible for marking the nodes for this verification.\nInformally, to perform the verification, each node must possess some information regarding each of the fragments *F* containing it. The information regarding a fragment *F* contains the weight of the selected edge of the fragment as well as the fragment identity, hence, it can be encoded using *O*(log*n*) bits. (The fragment identity is needed to identify the set *O**F* of outgoing edges from *F*, and the weight of the selected edge is needed for comparing it to the weight of the other edges of *O**F*; this is how we detect that the selected edge is indeed the minimum). However, as mentioned, each node participates in *O*(log*n*) fragments, and hence, cannot hold at the same time all the information relevant for its fragments. Instead, we distribute this information among the nodes of the fragments, in a way that will allow us later to deliver this information efficiently to all nodes of the fragment. In this section, we show how to distribute the information regarding the fragments. In the next section, we explain how to exploit this distribution of information so that during the verification phase, relevant information can be delivered to nodes relatively fast and without violating the *O*(log*n*) memory size.\n#### The piece of information `ID`(*F*):\nAs mentioned in Section [intuition:P2], a crucial point in the scheme is letting each node *v* know, for each of its incident edges (*v*, *u*) ∈ *E* and for each level *j*, whether *u* and *v* share the same level *j* fragment. (Note, in the particular case where *u* is also a neighbour of *v* in *T*, this information can be extracted by *v* using *u*’s data-structure, see Lemma [lem:simple-proof].) Intuitively, this is needed in order to identify outgoing edges. For that purpose, we assign each fragment a unique identifier, and *v* compares the identifier of its own level *j* fragment with the identifier of *u*’s level *j* fragment. The identifier of a fragment *F* is `ID`(*F*) : = `ID`(*r*(*F*)) ∘ *l**e**v*(*F*), where `ID`(*r*(*F*)) is the unique identity of the root *r*(*F*) of *F*, and *l**e**v*(*F*) is *F*’s level. We also need each node *v* to know the weight *ω*(*F*) of the minimum outgoing edge of each fragment *F* containing *v*. To summarize, the *piece of information* `I`(*F*) required in each node *v* per fragment *F* containing *v* is `I`(*F*) : = `ID`(*F*) ∘ *ω*(*F*). Thus, `I`(*F*) can be encoded using *O*(log*n*) bits.\nWhen describing the construction of hierarchy $\\cH\\_{\\cM}$ below, we view it as a tree (and call it the *fragment-tree*) whose nodes are fragments of *T*, and in particular, the root of the fragment-tree is *F*ℓ, the level ℓ fragment which is the whole tree *T*. We shall make sure that ℓ = *O*(log*n*). We refer to the nodes of the fragment-tree $\\cH\\_{\\cM}$ as *fragment nodes*. Consider a fragment node *F*, corresponding to a fragment of *T*. Whenever no ambiguity arises, we refer to *F* sometimes as a node of $\\cH\\_{\\cM}$ and sometimes as a fragment of *T*. For example, we may refer either to *F*ℓ or to *T*, whichever simplifies the notations better.\n### Constructing hierarchy $\\cH\\_{\\cM}$ and candidate function *χ*\n.\nUnfortunately, we did not manage to utilize one of the known decompositions of a tree, or an MST. One famous such decomposition is the set of fragments created in the algorithm of Gallager, Humblet, and Spira as building blocks of the MST in constructed by their seminal algorithm. First, many of their fragments are very large. We could not spread the fragment’s `I` over such a large fragment, since this would take the information far beyond the desired locality radius. Our construction below groups such fragments into an object we there term “phase 1” and replicate `I` over phase 1 as needed in order not to violate the locality radius constraint. The rest of the fragments are grouped into phases as well. This is motivated by our intention to show later (in Section [sec:distributed-implementation]) a distributed implementation of the labeling schemes presented here. Intuitively, the implementation needs to use some memory per an object, and let the marker algorithm distribute this memory over the nodes of the object (we shall see the details in Section [sec:distributed-implementation]). Had there been too many objects, we would have violated the memory constraint. Hence, we needed to group fragments together, leading the to “phases” structure described below.\nThe marker constructs the fragment-tree $\\cH\\_{\\cM}$ in *O*(log\\**n*) iterations. Each iteration constructs a forest (a collection of subtrees of $\\cH\\_{\\cM}$) termed *phase*. Specifically, each iteration *i* ≥ 1 that constructs a phase *P*, starts with a collection of fragment nodes, which then become the roots of the subtrees of phase *P*. We term them the *root fragments* of phase *P*. For an iteration *i* > 1, each such root fragment (of phase *P*) is a child (in $\\cH\\_{\\cM}$) of a fragment leaf of the phase *P*− constructed in the previous iteration. The first iteration starts with the root fragment of $\\cH\\_{\\cM"},"data_id":{"kind":"string","value":"arxiv_0182774"}}},{"rowIdx":45695,"cells":{"content":{"kind":"string","value":"Reinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging\n======================================================================================\n=1\nWe consider two data driven approaches, Reinforcement Learning (RL) and Deep Trajectory-based Stochastic Optimal Control (DTSOC) for hedging a European call option without and with transaction cost according to a quadratic hedging P&L objective at maturity (“variance-optimal hedging” or “final quadratic hedging”). We study the performance of the two approaches under various market environments (modeled via the Black-Scholes and/or the log-normal SABR model) to understand their advantages and limitations. Without transaction costs and in the Black-Scholes model, both approaches match the performance of the variance-optimal Delta hedge. In the log-normal SABR model without transaction costs, they match the performance of the variance-optimal Barlett’s Delta hedge. Agents trained on Black-Scholes trajectories with matching initial volatility but used on SABR trajectories match the performance of Bartlett’s Delta hedge in average cost, but show substantially wider variance. To apply RL approaches to these problems, P&L at maturity is written as sum of step-wise contributions and variants of RL algorithms are implemented and used that minimize expectation of second moments of such sums.\nIntroduction\n============\nData-driven hedging has attracted wide and deep interest as an application of reinforcement learning (RL) and similar approaches to derivatives pricing and risk management, see for instance,, or. These approaches have advantages, such as (1) that they can work with observed, simulated, or generated data for the behavior of the instruments used for hedging and risk management, rather than proceed in model-specific fashion, (2) that the value and other characteristics of a trading strategy over time can be easily modeled given such data, even for realistic models of incomplete markets where trades cause friction and attract transaction costs, and (3) that hedging or risk management measures of such strategies can be easily computed and used as objectives to be optimized over or controlled for. These measures could quantify how the strategies perform as measured against book-keeping or market prices of the to-be-hedged instruments or quantify the profits and losses when following that strategy to hedge or risk-manage the to-be-made or to-be-received payoffs for the to-be-hedged portfolios. In the later case, the behavior of such strategies can be modeled and optimized over even as consistent prices or sensitivities might be difficult or impossible to compute in conventional models with conventional methods. This allows the training of data-driven agents that optimize said hedging and risk management measures and that perform as well as or better than conventional sensitivy-based and model-based hedging approaches. Data-driven agents can also take other factors into account and automate or support hedging in situations where conventional hedging approaches require ad-hoc solutions and tweaks by traders and risk managers.\nSeveral banks and other players in the financial industry have started to investigate and apply these new approaches to hedging and risk management, such as reported in. However, the resulting agents, their behavior, and associated risks of use, including model risk, need to be studied and understood, so that appropriately designed and understood data-driven hedging and risk-management solutions can be applied safely and successfully. Here, we will study how RL and Deep Trajectory-based Stochastic Optimal Control (DTSOC) approaches can be applied to hedging problems, in particular hedging to control the quadratic deviation (variance) of the final profit-and-loss after having made and received all payoffs of the instrument or portfolio to be hedged (“quadratic final hedging”).\nScope of This Work\n------------------\nThis paper concentrates on hedging strategies that control the quadratic deviation (variance) of the final profit-and-loss after having made and received all payoffs of the instrument or portfolio to be hedged (“quadratic final hedging”). The setting will apply to other trading strategies with other objectives also.\nTo define the setting, we will model how the prices of the hedging instruments evolve, how the trading strategy to do so performs (allowing transaction costs), and define the final objective to be optimized over in terms of the prices, costs, etc. involved in the strategy.\nTo set up training of agents for final quadratic hedging, one only needs to model the prices of the hedging instruments, the trading strategy, and the payoffs of the to-be-hedged instrument or portfolio. This is different from the setting considered in, where one also needs to model the prices of the to-be-hedged instruments either with a consistent model or a book-keeping model. In situations where the to-be-hedged instrument and closely related instruments are not liquidly quoted nor traded (either on exchange or on some OTC system) and where no appropriate and flexible enough book-keeping models capturing the important features of the instruments and the trading (such as transaction costs) are available, final quadratic hedging is applicable (or anything that is defined purely in terms of a strategy involving the modeled prices of the hedging instruments) while step-wise mean-variance hedging would not be applicable.\nHere, we assume we have a model of how the appropriate risk factors and prices for the hedging instruments evolve as needed to compute the hedging objectives. This means that we have some implementation that can generate as many trajectories of these instrument prices and other risk factors as needed. Based on this data, one can simulate the trading strategy and determine its performance and thus the objective we will optimize over. It is assumed that the trading from the strategies does not impact how the risk factors and the prices of the hedging instruments evolve but only how the trading strategy and its results evolve, as is commonly assumed. We only assume that we can obtain trajectories as needed and do not need any information about how the model is obtained, setup, or run. Thus, either a well-specified traditional model could be used or some generative model trained on appropriate data as in or. We will work here with trajectories simulated from a Black-Scholes model or a log-normal SABR model while leaving synthetic data from generative models to future work.\nWe will test and validate our agents on simulated data from such models. Whether agents trained on simulated or generated data can perform well on observed data is an interesting research question and is left to future work. There is some work that shows that at least in some circumstances agents trained on simulated or synthetic data can perform well when run against observed market data.\nThe agents themselves are operating in a model-free, data-driven setting, since they only are provided generated trajectories and rewards rather than details about the model. Given our setting, we are in a rich data regime (even unlimited data regime), corresponding to a well defined conventional or generative model that is used to generated as much data as wanted.\nRelated Work\n------------\nWe discuss here some of the existing work in the literature on the use of Reinforcement Learning and Deep Trajectory Based Stochastic Optimal control to pricing and hedging of options. For some broader survey of RL applications in finance, see.\nIn, a European call option in the Black-Scholes model (trading at discrete times, no transaction costs) is hedged and priced using a *Q*-learning method (there, called QLBS method). The objective is minimizing the sum of the initial cash position and the weighted discounted sum of variances of the hedging portfolio at all subsequent time steps. This setting and objective can be identified with a corresponding MDP setup and thus solved with standard RL/DP approaches such as *Q*-learning. The price at maturity has to be equal to the final payoff and prices respective *Q*-functions at earlier times are determined by Dynamic Programming or other standard RL approaches.\n defines the objective as a combination of the expected final P&L and the expected variance of the final P&L. They use versions of *Q*-learning and other methods for the expected sum of contributions and the expected square of sums of contributions to address this problem. We will use similar approaches to optimize the variance of the final P&L of the hedged portfolio, but did not discuss or apply their approaches to final quadratic hedging. test their methods on data simulated from the Black-Scholes and log-normal SABR model and compare their results to Delta hedging with current implied volatility and Bartlett’s Delta hedging with current implied volatility.\nMany works consider neural networks as a numerical tool to accelerate calculation tasks and to solve generic classes of problems, such as model calibration, solving PDEs, solving stochastic optimal control problems more efficiently and in higher dimensions. Among these, we distinguish the line of works initiated in where neural networks are used to solve certain stochastic optimal control problems numerically. More concretely, the DTSOC method in is a setup for trajectory based empirical deep stochastic optimal control with both step-wise and final objectives/costs. Finally, the “deep hedging” paper presents a trajectory based empirical deep stochastic optimal control approach to minimizing some global objectives related to replication and/or risk management of some final payoff, according to risk measures such as CVaR/expected shortfall, expected utility, etc. While quadratic hedging is mentioned, the method(s) are not applied to any variance-optimization/quadratic hedging problem in that paper.\nFramework\n=========\nGeneral Setup\n-------------\nWe consider here dynamic hedging, trading, and/or risk management problems defined in terms of some instrument or portfolio to be hedged or otherwise risk managed. In our examples, we assume that the instruments are European options; but one could handle portfolios or such or more path-dependent instruments in a very similar fashion. In these problems, there is an universe of given instruments (including cash/bank accounts that allow also negative balances) and the agent has to decide how much of each instrument to hold at each time to achieve a certain hedging, trading, or risk management objective. We assume, as is commonly done, that there is a given fixed set of times at which trades can occur and we allow that such trades might incur transaction costs. We also assume that the strategies are self-financing. The objectives under consideration can be defined from contributions at one or several fixed time horizons or each trading period and/or trading time might have a contribution associated to it. Here, we will consider objectives that are associated with the one final payment time of the to-be-hedged instrument. While the objective can be a simple mean-square hedging error as here, it could also be defined based on a risk measure (such as CVaR) or some moments or percentiles of an appropriately defined hedging outcome.\nThis is a stochastic sequential decision problem. The prices of the hedging instruments (here, stocks) will follow some stochastic process with some given model or generator. These prices can be observed and other appropriate risk factors could be computed based on the history of such prices. Now, given these observed prices and computed risk factors, the agent needs to decide in time for each trading time how much of each hedging instrument to buy and/or sell; or, equivalently, how much of each hedging instrument should be held immediately after that trading time; taking into account transaction costs and other trade impacts. (Thus, holding sizes or trade sizes would be appropriate decision variables.) Given to-be-achieved holding size or chosen trade size and the cash at hand before the trade, the cash remaining after (or being borrowed) can be computed by self-financing and funding strategy, the value of the holdings and of the cash at the end of the no-trade interval follows from the stochastic evolution of the prices of the hedging instruments and the interest on the cash or loan account, up to final maturity.\nTrading Strategy for the Hedging Portfolio\n------------------------------------------\nWe consider the case of a hedging portfolio. This means that we are given some description of the to-be-hedged instrument or portfolio. This description in general consists of random variates (*P**t*)*t* ∈ T describing the contractually-agreed payoffs under that portfolio, where T is the set of their payment times. We assume here that we are treating European options: there is only one payoff at some maturity *T* which we denote *P**T*.\nAs in, it is assumed that we are given “hedging” instruments (say, one or several stocks), we denote their prices by *S**t* (in general a vector), and denote the holding of them in our strategy at time *t* with *H**t**S* (a vector of corresponding length). For our examples, we will typically assume a single hedging instrument. Finally, we assume that excess cash from the trading strategy is deposited in a money-market account with a certain given interest accrual and discount factor (and that negative cash balances can be borrowed at the same interest accrual and discount factor). We assume that we initially start with some initial cash *Y*0 but no holdings in the hedging instruments (i.e., *H*0−*S* = 0). The trading times are given as *t**i* with *t*0 = 0. If referred to, *t*− 1 is a time just before time 0 with no holding in hedging instruments.\nThen, at each time *t**i* when one needs to decide what to trade, one can observe the prices *S**t**i* of the hedging instruments. At the same time the holdings in these instruments are still given by *H**t**i* − 1*S*. Now one has to decide how much (*H**t**i**S*) shall be held for the next no-trading period. Here, we assume that these trades are made instantaneously at *t**i*+. The minimal state *s**i* for this decision has to include at least (*t**i*, *S**t**i*, *H**t**i* − 1*S*).[2](#fn2) The action has to either describe *H**t**i**S* or how to obtain *H**t**i**S* from *H**t**i* − 1*S*.\nConsider time *t**i* + 1, right before the trade: The value of the hedging part of the strategy then is \n \n$$Y\\_{t\\_{i+1}} = \\frac{\\left( Y\\_{t\\_{i}} - H^S\\_{t\\_i} S\\_{t\\_i} - \\mathsf{TC}(H^S\\_{t\\_i},H^S\\_{t\\_{i-1}},S\\_{t\\_i}) \\right)} {\\mathsf{DF}\\_{i,i+1}} + H^S\\_{t\\_i} S\\_{t\\_{i+1}}$$ \nThis reflects that at the previous trade time *t**i*, the portfolio was rebalanced to the appropriate holdings *H**t**i**S*, transaction costs were charged TC(*H**t**i**S*, *H**t**i* − 1*S*, *S**t**i*), the thus resulting cash balance *Y**t**i* − *H**t**i**S**S**t**i* − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S**t**i*) attracts step-wise interest (interest accrual corresponds to dividing by appropriate step-wise discount factor *D**F**i*, *i* + 1). At the same time, the holdings in the hedging instruments are now valued based on the new price of the hedging instruments.\nIt helps to rewrite the equation as follows as in :\n \n$$Y\\_{t\\_{i+1}} = \\frac {1} {\\mathsf{DF}\\_{i,i+1}} Y\\_{t\\_i} + H^S\\_{t\\_i} \\left( S\\_{t\\_{i+1}} - \\frac {S\\_{t\\_i}} {\\mathsf{DF}\\_{i,i+1}} \\right)\n- \\frac{\\mathsf{TC}(H^S\\_{t\\_i},H^S\\_{t\\_{i-1}},S\\_{t\\_i})} {\\mathsf{DF}\\_{i,i+1}}$$ \nIn this completely linear setting without stochastic interest rates (and assuming that the transaction cost is linear in the instrument prices for positive multipliers *c*, TC( ⋅ , ⋅ , *c**S*) = *c*TC( ⋅ , ⋅ , *S*)), it is possible to “hide” the discounting in the value definitions in the following sense: with *Ỹ**t**i* = DF0, *i**Y**t**i* and *S̃**t**i* = DF0, *i**S**t**i*, we have\n \n*Ỹ**t**i* + 1 = *Ỹ**t**i* + *H**t**i**S*(*S̃**t**i* + 1 − *S̃**t**i*) − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S̃**t**i*) \nIn particular, this means for the increment\n \n$$\\tilde{dY}\\_{t\\_{i+1}} = H^S\\_{t\\_i} \\left( \\tilde{S}\\_{t\\_{i+1}} - \\tilde{S}\\_{t\\_i} \\right)\n- \\mathsf{TC}(H^S\\_{t\\_i},H^S\\_{t\\_{i-1}},\\tilde{S}\\_{t\\_i})$$ \nand we thus do not get any contribution or impact of the initial cash position *Y*0 beyond an additive shift.\nFinal Quadratic Hedging Objective\n---------------------------------\nWe will also discount the final payoff and denote it *P̃**T*. The discounted final hedging error is now:\n \n*L̃**T* = *P̃**T* − *Ỹ**T* \nand we look to minimize the mean square hedging error MSHE (“quadratic hedging”):\n \nMSHE(*Y*0, *H*⋅*S*) = E[*L̃**T*2] \nwhere the expectation is under the given model respective generator for the underlying hedging instruments. This hedging is also referred to as variance optimal hedging (where the variance is of the final hedging error). We typically avoid this name except when referring to the literature since it does not further specify which variance and use “final quadratic hedging” since this makes it clearer that we optimize over the expectation of the square of the final hedging mismatch.\nExtensions to stochastic interest rates and discounting, differential rates, more involved funding policies are possible, require further notation and details, and can be handled similarly, but are not necessary for the settings we discuss here. Differential rates and more involved funding policies could be included in the framework by adding bank accounts for positive and negative balances and other funding instruments to the hedging instruments and adding constraints to the decision variables (only positive holdings of positive bank account, only negative holdings of negative bank account i.e. bank debits or loans, restricting secured loans/repos to the amount held in the corresponding collateral, etc.).\nUnder some circumstances, the strategy might spend a lot of transaction costs to decrease the final hedging error (and thus requiring larger initial wealth *Y*0). In such cases, one might constrain *Y*0 or also minimize over initial wealth, or one might constrain total transaction cost - or also minimize over accumulated total transaction costs.\nSince the impact of initial wealth is only additive and separated, we can compute the optimal *Y*0 for each hedging strategy *H*⋅*S*. The formulation also covers the case in which *Y*0 has been chosen from the outside (prices set by other mechanism or the instrument is on the run and we start with the existing cash holdings (and/or stock holdings) assigned to the hedging of our portfolio) where we only optimize over the strategy for the remaining time until maturity.\nModeling the Hedging Instruments\n--------------------------------\nThe formulation of the objective function in this paper only depends on the final hedging error - i.e., the difference between the contractually agreed (discounted) payoff *P̃**T* and the final discounted value of the hedging strategy *Ỹ**T*. *Ỹ**T* in turn depends on the discounted prices of the hedging instruments and transaction costs earlier, but does not require any further information about the models for these instruments and prices.\nThe agent only needs to determine how many units to hold in each of the hedging instruments after rebalancing at each rebalancing time *t**i*, i.e. *H**t**i**S*. Alternatively, one could specify the units to be sold or bought at any given time together with the initial position; or assume that the amount sold or bought is proportional to the time between trades or some other variable and then specify that proportional trading rate in that setting. The minimal state for modeling this decision problem are these discounted prices (hedging instruments), the amount of the hedging instruments, time, and whatever state the models for these prices and the transaction costs need.\nAs for the models for the prices of the hedging instruments, one can model them under pricing or observational measure, with some differences in parameters and calibration or fitting. For the hedging instruments, one can use conventional quantitative finance models, possibly with hidden factors, such as Black-Scholes, Local Volatility Model, SABR model (with stochastic volatility), Heston model (with stochastic variance), quadratic rough Heston model, etc. One could use generative models such as GANs or appropriately trained neural Stochastic Differential Equations (SDEs) or similar. Finally, one could use historically observed data over one instrument or a cross section of similar instruments together with appropriate assumptions to generate possible future price movements or future prices.[3](#fn3)\nFor the to-be-hedged instrument, we only need to know the payoff at maturity and we do not need any model or method to compute the price of the to-be-hedged instrument at any other time, unlike in hedging and risk management setups that require book-keeping or reference prices for the to-be-hedged instruments. We consider at first simple examples such as the hedging of a short call option that was sold to some counterparty or the hedging of a long call option that was bought. Here, one only needs to specify the final payoff *P**T* = − (*S* − *K*)+ respectively *P**T* = (*S* − *K*)+. Black-Scholes models with constant or deterministic volatility and log-normal SABR model are considered here, but other models can be easily implemented.\nIn formulas, this would mean for the discounted price of the hedging instruments (“stocks”) in continuous time\n \n*d**S̃**t* = *σ**t**S̃**t**d**W**t* \nwith *σ**t* a constant or deterministic time-dependent function (Black-Scholes), or following the stochastic volatility SDE of a SABR model\n \n$$d\\sigma\\_t = \\frac{\\eta}{2} \\sigma\\_t dW^2\\_t$$ \nand in discrete time, appropriate log-Euler time-discretizations thereof (which are actually exact for *S̃**t* in this case). The Black-Scholes SDE also corresponds to the SDE for the underlier in a log-normal SABR model (with *β* = 1).\nIf one considers more complicated models based on conventional quantitative finance models, one would simulate the (discounted) underlying instruments with that model. If these models have additional factors, these factors would be added to the state. If some of these factors are latent or hidden factors, one would need to add some mechanism how these latent factors can be estimated or taken into account by some process on observed quantities, add these observed quantities to the state, and learn agents that only depend on observed and observable quantities, not the latent factors that will in general be unknown (and unknowable) to the agent. The stochastic volatility in our log-normal SABR test case can be considered either as a latent, unobserved state or as observed state. (We treat it here as a latent state that is not provided as input to the agent/policy.)\nIn general, trading in hedging instruments could impact the prices in those hedging instruments either temporarily or permanently. We assume here that the hedging instruments are traded liquidly and that the hedged instrument is such that it can be hedged without impacting the prices of the hedging instruments. To a certain extent, short-term price impact can be modeled by and absorbed into the transaction cost terms.\nAs discussed in abstract and introduction, here we focus on Black-Scholes and the log-normal SABR stochastic volatility models to investigate the agents and algorithms in a setting where the model and the features are simple enough, and will consider other models in future work.\nAnalytical Exact or Approximate Variance-Optimal Hedges\n=======================================================\nFor some models (Black-Scholes, SABR, Rough Bergomi) and some assumptions (time-continuous trading, accurate enough approximations), one can derive explicit analytical forms of exact or approximate variance-optimal hedges that minimize MSHE under those assumptions. derives the below formulas (for the case of zero interest rates) and applies them to the log-normal SABR and the Rough Bergomi model.\nAssume thus that we have given *C**t* as the time *t* price for an instrument with payoff *P**T* (which means in particular that *C**T* = *P**T*) and denote the discounted-to-time-zero price as *C̃**t* = *D**t**C**t*, with *D**t* being the time zero discount factor for time *t*, assuming also deterministic interest rates. Similarly denote the discounted-to-time-zero version of *S**t* by *S̃**t* = *D**t**C**t*. We assume that we simulate and operate under a pricing measure Q under which *C̃**t* and *S̃**t* are square integrable martingales. Given any dynamic continuous-time trading strategy *θ*⋅, the final discounted value of the hedging portfolio if started from initial wealth *w* would be\n \n*Ỹ**T* = *w* + ∫0*T**θ**u**d**S̃**u* \nThe final hedging error is thus\n \n*L̃**T* = *C̃**T* − *Ỹ**T*. \nWe minimize over the (risk-neutral) mean-square hedging error\n \nMSHE(*w*, *θ*⋅) = E[*L̃**T*2] \nand the minimizer under the above assumptions is called variance-optimal strategy *θ**V**O*. It is given as the Radon-Nikodym derivative of the finite-variation process ⟨*S̃*, *C̃*⟩*t* with respect to the finite-variation process ⟨*S̃*, *S̃*⟩*t*:\n \n$$\\theta^{VO}\\_t = \\frac {d\\langle\\tilde{S},\\tilde{C}\\rangle\\_t} {d\\langle\\tilde{S},\\tilde{S}\\rangle\\_t}.$$ \nFurther assume that *P**T* represents a call (similar derivations work for puts) and that we write its discounted price in terms of implied volatility Σ*t* by the Black-Scholes formula for zero interest rates (here denoted by *c**B**S*)[4](#fn4):\n \n*C̃**t* = *c*BS, *t*(*S̃**t*, Σ*t*; *T*, *K̃*) = *c*BS, *t*(*S̃**t*, Σ*t*). \nThis Σ*t* will be a stochastic process (and as a smooth function of martingales, will be a semi-martingale). We introduce\n \n$$\\begin{aligned}\n d\\_t^{\\pm} (\\tilde{S}\\_t,\\Sigma\\_t) &=&\\frac{\\log(\\tilde{S}\\_t/\\tilde{K})}{\\Sigma\\_t\\sqrt{T-t}}\\pm\\frac{\\Sigma\\_t\\sqrt{T-t}}{2}\\\\\n \\mathsf{Delta}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)&=&\\Phi(d\\_t^{+}(\\tilde{S}\\_t,\\Sigma\\_t))\\\\\n \\mathsf{Vega}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)&=&\\tilde{S}\\_t\\phi(d\\_t^{+}(\\tilde{S}\\_t,\\Sigma\\_t))\\sqrt{T-t}.\\end{aligned}$$ \nIn terms of Σ*t*, we obtain:\n \n$$\\label{QH\\_optimal\\_formula2}\n\\theta^{VO}\\_t= \\mathsf{Delta}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)\n+\\mathsf{Vega}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)\\frac{d\\langle\\Sigma,\\tilde{S}\\rangle\\_t}{d\\langle\\tilde{S},\\tilde{S}\\rangle\\_t}$$ \nwith the corresponding Radon-Nikodym derivative in the second term.\nNote that if Σ*t* is constant or a deterministic function of time, the second term is zero and Delta*t* is a variance optimal strategy, as in Black-Scholes with constant or at most time-dependent volatility *σ*(*t*).\nThus, if we have an explicit Σ*t* and we can compute *d*⟨Σ, *S̃*⟩*t* and *d*⟨*S̃*, *S̃*⟩*t* explicitly, we have an explicit variance optimal strategy. We can also use close approximations $\\hat{\\Sigma}\\_t$ of Σ*t* and would expect the resulting *θ**t**A**V**O* to be a good approximation of *θ**t**V**O*.\nLog-normal SABR for the discounted-to-time-zero stock price *S̃**t* is written\n \n$$\\begin{aligned}\nd\\tilde{S}\\_t=&\\tilde{S}\\_t\\sigma\\_t dW\\_t\\\\\nd\\sigma\\_t=&\\frac{\\eta}{2}\\sigma\\_t dW^{2}\\_t,\\end{aligned}$$ \nwhere *d*⟨*W**t*, *W**t*2⟩*t* = *ρ**d**t* for *ρ* ∈ [ − 1, 1], we have\n \n$$\\begin{aligned}\n\\hat{\\Sigma}\\_t & = & \\sigma\\_t f(M\\_t) \\\\\nM\\_t &=& \\frac{\\eta}{\\sigma\\_t} \\log \\left( \\frac{\\tilde{K}}{\\tilde{S}\\_t} \\right)\\end{aligned}$$ \nwith *f* given in the SABR formula ( and ):\n \n$$f(y) = \\frac{y/2}{\\log\\left(1-\\rho\\right) - \\log\\left(\\sqrt{1+\\rho y+y^2/4} - \\rho - y/2\\right)}$$ \nThe (approximated) Radon-Nikodym derivative can be computed as follows :\n \n$$\\frac{d\\langle\\hat{\\Sigma},\\tilde{S}\\rangle\\_t}{d\\langle\\tilde{S},\\tilde{S}\\rangle\\_t} = \\frac{\\eta}{2\\tilde{S}\\_t} \\left( \\rho f(M\\_t) - (\\rho M\\_t+2)f'(M\\_t) \\right)$$ \nThus, we obtain an approximately variance optimal strategy:\n \n$$\\theta^{AVO}\\_t= \\mathsf{Delta}\\_t(\\tilde{S}\\_t,\\hat{\\Sigma}\\_t)+\\frac{\\eta}{2} \\phi(d\\_t^{+}(\\tilde{S}\\_t,\\hat{\\Sigma}\\_t))\\sqrt{T-t} \n\\left( \\rho f(M\\_t) - (\\rho M\\_t+2)f'(M\\_t) \\right)$$ \nThis turns out to be exactly Barlett’s delta in this case (*β* = 1). A similar approximation also exists for the one-factor Rough Bergomi model (which includes the Brownian Bergomi model as a special case).\nWhile our hedging problem is posed with discrete trading times and transaction costs, these exact or approximate explicit analytical strategies should be close to optimal for frequent trading without transaction costs in the parameter domains where the approximation performs very well. Otherwise, it should certainly serve as a useful reference point.\nDecision processes\n==================\nSettings\n--------\nMany problems such as the ones considered here can be cast in the context of an agent interacting with some environment based on some decision policy that tries to optimize (maximize or minimize, as the case may be) some function of a sum of step-wise, initial, and final contributions (“rewards”).[5](#fn5)\nTo specify abstractly such settings, one needs to specify the state of the environment, the set of possible inputs to the agent, and the set of possible actions by the agent. One also needs to specify how the environment transitions to a new state with new observations in reaction to an action (and this transition can be random and parts of the state might not be affected by the action) and what contribution each action and step will make to the sum of rewards (and that contribution might be stochastic as well).\nIn some situations, the state and the evolution of the environment might not be completely captured. This makes such a setting very hard to handle and standard results do not apply and we will not discuss such settings further. The transitions and rewards might potentially depend on all the states and actions that came before, presenting Nonmarkovian problems. Most algorithms and theory are presented for the Markovian case where both transitions and rewards only depend on the most recent previous state and the current action, and we will only discuss this setting here. In at least some cases it is possible to exactly or approximately capture path dependency by extending the state space (such as for up or down Barrier options keeping track of running maximum or minimum or whether the corresponding barrier has been breached yet or not) and then using Markovian methods. There are circumstances under which the agent only observes some part or function of the state etc., called partially observable decision processes. While such settings might be needed to treat settings in which there are latent parts of the state (such as an unobserved stochastic volatility), we will not cover their setting or theory here.\nOne sometimes introduces a reward discount factor *γ* ∈ (0, 1] if optimizing earlier rewards or costs is more important than optimizing later contributions, with *γ* = 1 corresponding to the undiscounted case. (One could potentially allow *γ* ≥ 1 to put more emphasis on later contributions.) Such *γ* also might help with the convergence or numerical behavior of the sum but a *γ* not equal to 1 does change the problem and its solution in general. In our setting, we are mostly interested in the *γ* = 1 case and might look at values close to 1 for numerical purposes.\nMarkov decision processes (MDPs) with deterministic rewards can be specified by the following:\n* *state space* S: describes the state of the evolution of the environment and what the agent takes as input to make a decision\n* *action space* A: describes the set of agent actions\n* *Transitions*:\n\t+ *Transition probability* P : S × A × S → [0, 1] or *transition density* *p* : S × A × S → R+: P(*s*, *a*, *s*ʹ) denotes the probability of the state transitioning from *s* to *s*ʹ as a result of taking the action *a*, and the next state *s*ʹ would be chosen based on the probability P(*s*, *a*, ⋅ ), or\n\t+ *Transition function* with explicit randomness: *s*ʹ = *f*(*s*, *a*, *ξ*) with *ξ* describing the randomness and stochastic impact that impacts the transition between *s* and *s*ʹ.\n* Deterministic *rewards* *r* : S × A × S → R: reward assigned to the transition (*s*, *a*, *s*ʹ).\n* *γ*: the *reward discount factor*.\nThe set of trajectories or realizations of MDP can be described as follows. One first starts with an initial state *s*0 that could be fixed or given as a random variate or otherwise varying. Then, starting with time index *t* = 0, *s**t* is given, agent chooses an action *a**t*, and environment generates a new state *s**t* + 1 = *f*(*s**t*, *a**t*, *ξ**t*) and a reward *R**t* + 1 = *r*(*s**t*, *a**t*, *s**t* + 1). Since there is randomness at least in the transition to the new state, *s**t* + 1 and *R**t* + 1 will be (at least partially) random and their distributions will depend on the actions taken. The agent’s decisions are generated from a policy *π*. A deterministic policy is a function *π* : S → A and a stochastic policy assigns a distribution *π*(.∣*s**t*) over the set of actions where actions are drawn from *a**t* ∼ *π*(*s**t*).\nE*π*[.] = E*a**t* = *π*(*s**t*)[.] respectively E*π*[.] = E*a**t* ∼ *π*(.∣*s**t*)[.] denote the expectation under the randomness of the state transition if the actions *a**t* of the agent are chosen according to policy *π* for deterministic or stochastic policies, respectively.\nThe *return* of a MDP starting at time index *t* is defined as\n \n*G**t* = ∑*k* = 0∞*γ**t* + *k**R**t* + *k* + 1, \nThe return satisfies the recursion, *G**t* = *R**t* + 1 + *γ**t* + 1*G**t* + 1.\nValue functions and action-value functions\n------------------------------------------\nFor a fixed policy *π*, the *value function* assigns to each state the expected return, conditional on starting from state and selecting actions according to policy *π* thereafter,\n \n*V**π*(*s*) = E*π*[*G**t*∣*s**t* = *s*]. \nMarkov property gives a fundamental recursive structure to the value function, the Bellman equation for the value function:\n \n$$\\begin{aligned}\nV^{\\pi}(s) = R\\_{t+1}+\\gamma\\sum\\_{s'}\\mathbb{P}^{\\pi}(s,s')V^{\\pi}(s'), \\label{bellman\\_eq}\\end{aligned}$$ \nwhere, P*π*(*s*, *s*ʹ) = P(*S**t* + 1 = *s*ʹ∣*s**t* = *s*, *a**t* ∼ *π*(*s**t*)).\nThe return of an MDP is a random variable, so one can take (conditional) expectations of its second moment:\n \n$$\\label{return\\_2nd\\_moment}\nM^{\\pi}(s) \\coloneqq \\mathbb{E}\\_{\\pi}\\left[G\\_t^2|s\\_t=s\\right].$$ \nIt is possible to formulate a Bellman equation for the expectation of the second moment of the return (derived in for a discounted infinite horizon with finite states and for the finite horizon in ) as follows: one has the expansion\n \n$$\\begin{aligned}\nG\\_t^2 &= (R\\_{t+1}+\\gamma G\\_{t+1})^2\\\\\n &= R^2\\_{t+1}+2\\gamma R\\_{t+1}G\\_{t+1}+\\gamma^2 G\\_{t+1}^2.\\end{aligned}$$ \nTherefore, one can re-write the expectation of the second moment of the return as,\n \n$$\\begin{aligned}\nM^{\\pi}(s) &= \\mathbb{E}\\_{\\pi}\\left[G\\_t^2|s\\_t=s\\right]= \n\\mathbb{E}\\_{\\pi}\\left[R^2\\_{t+1}+2\\gamma R\\_{t+1}G\\_{t+1}+\\gamma^2 G\\_{t+1}^2\\right]\\\\\n &= \\mathbb{E}\\_{\\pi}\\left[R^{(M)}\\_{t+1}\\right]+ \n \\mathbb{E}\\_{\\pi}\\left[\\gamma^2 M^{\\pi}(s\\_{t+1})\\right],\\end{aligned}$$ \nwhere,\n \n*R**t* + 1(*M*) = *R**t* + 12 + 2*γ**R**t* + 1*G**t* + 1. \nBy defining,\n \n$$r\\_{\\pi}^{(M)}(s) \\coloneqq\\mathbb{E}\\_{\\pi}\\left[R\\_{t+1}^{(M)}|s\\_t=s\\right]$$ \nthe Bellman equation for the expectation of the second moment takes the form,\n \n*M**π*(*s*) = *r**π*(*M*)(*s*) + ∑*s*ʹP*π*(*s*, *s*ʹ)*γ*2*M*(*s*ʹ). \nWe introduce another concept, closely related to the value function. The action-value function or the *Q-function* at state *s* and action *a* is the value of taking the action *a* and following the policy *π* after that,\n \n$$Q^{\\pi}(s,a) = \\mathbb{E}\\left[\\sum\\_{t=0}^T\\gamma^t R\\_{t+1}\\middle|s\\_0=s, a\\_0=a\\right].$$ \nThe value function can be expressed as the expectation of the Q-function across all possible actions,\n \n$$V^{\\pi}(s) = \\mathbb{E}\\left[Q^{\\pi}(s,a)\\middle|a\\_t\\sim\\pi(.|s)\\right].$$ \nThe Q-function satisfies the Bellman expectation equation,\n \n$$\\label{Q\\_Bellman\\_equation}\nQ^{\\pi}(s,a) = \\mathbb{E}\\left[ R\\_{t+1} + \\gamma Q^{\\pi}(s', a') \\;\\middle|\\; s\\_t = s, a\\_t = a\\right].$$ \nStandard reinforcement learning tries to find an optimal policy that maximizes the expected return, *π*\\* , *s**t**d* which would give the maximum action-value function *Q*(\\**s*, *a*) for any *s* and *a*,\n \n*Q*\\*(*s*, *a*) = *Q**π*\\* , *s**t**d*(*s*, *a*) ≥ *Q**π*(*s*, *a*). \nThere is a version of Bellman equation for the *Q*\\*-function:\n \n$$Q^{\\ast}(s,a) = \\mathbb{E}\\left[ R\\_{t+1} + \\gamma \\max\\_{a'} Q^{\\ast}(s', a') \\;\\middle|\\; s\\_t = s, a\\_t = a\\right]. \\label{bellman\\_optimality\\_equation}$$ \nSimilar to ([return2ndmoment]), one can define an action-value function for the expectation of the second moment as:\n \n$$K^{\\pi}(s,a) \\coloneqq \\mathbb{E}\\_{\\pi}\\left[G\\_t^2|s\\_t, a\\_t\\right]$$ \nand write that out as,\n \n$$\\label{second\\_moment\\_q}\nK^{\\pi}(s,a) = \\mathbb{E}\\left[ R\\_{t+1}^2 + 2\\gamma R\\_{t+1}Q^{\\pi}(s', a') + \\gamma^2 K^{\\pi}(s', a') \\;\\middle|\\; s\\_t = s, a\\_t = a\\right].$$ \nAs expected, the expectation of the second moment depends on that of the first moment. Therefore, to achieve a good approximation of the expectation of the second moment, a good estimate of the expectation of the first moment, i.e. the Q-function, is required. In section [ddpgsecond] we will use ([secondmomentq]) to find a policy that optimizes the second moment of Q-function. Similar ideas have been proposed by to optimize a combination of the return of the MDP and its variance.\nFinal quadratic hedging as Markov decision problem\n--------------------------------------------------\nOne way to fit quadratic hedging into an MDP/reinforcement-type learning set-up is to find such definitions of *γ* and *R**t* respective *R*(*s*, *a*, *s*ʹ) so that *G*0 = ∑*t* = 0*N**γ**t**R**t* + 1 = *L̃**T* or *G*0 = *L**T* (for deterministic interest rates, those two have the same optimizers). One typically uses *γ* = 1 or very close to 1 since *γ* cannot be independently chosen. One could try to learn strategies for smaller *γ* (or larger *γ*) and then move *γ* closer to 1. Some reinforcement algorithms only work (or can be proven to work) for *γ* < 1 and certain assumptions on rewards. We would then need a setting that satisfies those requirements. The methods we implemented here seem to work well enough for *γ* equal or very close to 1. Then, minimizing *G*02 will correspond to minimizing *L̃**T*2.\nWe thus set *γ* = 1, for *i* = 1, ⋯, *N* − 1 (with *T* = *t**N*)\n \n*R**i* = *H**t**i**S*(*S̃**t**i* + 1 − *S̃**t**i*) − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S̃**t**i*) \nand\n \n*R**N* = − TC(0, *H**T**S*, *S̃**T*) − *P**T*(*S̃**T*). \nIt is also possible to introduce some reference value function or process Ref with Ref*T* = *P**T*(*S̃**T*) and use it to “shift” rewards around:\n \n*R**i* = *H**t**i**S*(*S̃**t**i* + 1 − *S̃**t**i*) − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S̃**t**i*) + (Ref*t**i* − Ref*t**i* − 1) \nwhere Ref*t*− 1 = 0. Ref*t* will telescope out and lead to the same *G*0 and *G*02 (and same optimizers) but the formulation with Ref might be more amenable to particular reinforcement learning approaches. We did not apply this idea here since we were able to run the implemented reinforcement learning approach without introducing such a function or process which also means that one does not need to compute or define it.\nIn, ([cashflowform]) is called the cash flow formulation, while ([accountingform]) is called the accounting formulation.\nMethods\n-------\nStandard reinforcement learning uses Bellman equations, together with temporal difference learning, and/or policy gradient lemma, in various types of methods. There are model based approaches in which the specification of the transitions and rewards are used and the sums or integrals are computed exactly or with some approximation so that all (or, at least, many) transitions and rewards are taken into account at once. However, this is only possible in some cases, often requires a finite set of states, and will often taken considerable computing power and memory. There are also various model free approaches that work based on observed transitions and rewards (single or a few observed steps each) rather than entire trajectories. Reinforcement learning algorithms try to learn value and action-value functions and a policy (or using a greedy or *ε*-greedy policy derived from the action-value functions). For standard reinforcement learning, these methods are discussed at many places, we only point to and work cited therein as well as for a treatment and discussion of reinforcement learning approaches and stochastic optimization approaches for a variety of settings. However, there are many introductions to reinforcement learning available. We will discuss reinforcement-type learning approaches that optimize the expectation of the second moment in the next section.\nThere are methods that work based on empirical/MC full-trajectory simulation of the sum of rewards, parametrizing the agent’s policy as deep neural networks, and optimizing the policy by taking gradient or stochastic gradient steps (or extensions thereof, such as ADAM). We call these methods deep trajectory based stochastic optimal control (DTSoC) methods. In these methods, one does not (need to) learn value or action-value functions. We will discuss such methods in the section after the next.\nIf the evolution of the environment as a system of SDEs and the controlled SDE can be captured in forward-backward stochastic differential/difference equations (FBSDE) and the objective function has appropriate initial, step-wise, and/or final terms, one can apply deep learning methods for FBSDE, which need limited model information and/or outputs but are otherwise generic. We will discuss such methods in a later subsection.\nReinforcement Learning\n======================\nWe refer to for a description of Q-learning, policy gradient methods, and deep deterministic policy gradient methods, to maximize the return of the MDP with standard reinforcement learning. One can implement a variant of Q-learning that learns both *Q* and *K* functions. Here, we will present, implement, and use a variant of deep policy gradients that learns a *Q* function, a *K* function, and a policy that minimizes the variance, i.e., minimizes the *K* function.\nDeep Deterministic Policy Gradients Variant for K-function\n----------------------------------------------------------\nIn this section, we will propose an algorithm which instead of optimizing *Q*-function optimizes *K**π*(*s*, *a*), the variance of the return rather than the expected return. Our proposed algorithm involves the same ideas as DDPG to approximate the *Q*-function. However, there are two major differences between our algorithm and the original DDPG algorithm: 1) The original algorithm includes one network to approximate the *Q*-function for the policy and one network to approximate the policy and it updates the parameters of the *Q*-network by minimizing the mean squared error of the difference between its approximation and target values. However, our algorithm includes an extra network *K**ω*(*a*, *s*) with set of parameters *ω* to estimate *K**π*(*s*, *a*). We update the parameters *ω* by minimizing the mean squared error between the output of the *K**ω* and its target values. The target values for the K-network are different from target values of *Q*-network and are generated using ([secondmomentq]). 2) The policy in the original DDPG is updated toward choosing the actions that optimizes the Q-function of the policy at a specific state *s*, however, in our algorithm we update the policy to choose the action that optimizes (minimizes) the K-function - the expectation of the second moment of the return. The details of our method are provided in Algorithm [mainalg]. See for a similar method that minimizes a combination of *Q* (mean) and *K* (variance) functions.\n[mainalg]\nInitialize policy parameters *θ*, Q-function parameters *ϕ*, K-function parameters *ω* Set target parameters equal to main parameters *θ*targ ← *θ*, *ω*targ ← *ω*, *ϕ*targ ← *ϕ* Initialize replay buffer M Initialize a random process N for exploration Observe initial state *s*1 Select action *a**t* = *μ**θ*(*s*) + N*t* Execute action, receive reward *r**t* and state transition *s**t* + 1 Append the transition to the replay buffer M Sample a mini-batch of transitions *B* = {(*s**i*, *a**i*, *r**i*, *s**i* + 1)} from the replay buffer M target*Q* = *r* + *γ**Q**ϕ*targ(*s*ʹ, *μ**θ*targ(*s*ʹ)) target*K* = *r*2 + *γ*2*K**ω*targ(*s*ʹ, *μ**θ*targ(*s*ʹ)) + 2*γ**r**Q**ϕ*targ(*s*ʹ, *μ**θ*targ(*s*ʹ)) **Q-update:** Get updated *Q*-function network weights *ϕ*updated by one-step gradient descent,\n \n$$\\nabla\\_{\\phi}\\frac{1}{|B|}\\sum\\_{(s, a, r,s')\\in B}\\left(Q\\_{\\phi}(s,a)-\\text{target}\\_{Q}\\right)^2$$ \n**K-update:** Get updated *K*-function network weights *ω*updated by one-step gradient descent,\n \n$$\\nabla\\_{\\omega}\\frac{1}{|B|}\\sum\\_{(s, a, r,s')\\in B}\\left(K\\_{\\omega}(s,a)-\\text{target}\\_{K}\\right)^2$$ \n**Policy update:** Get updated policy network weights *θ*updated by one-step gradient descent,\n \n$$\\nabla\\_{\\theta}\\frac{1}{|B|}\\sum\\_{s\\in B}K\\_{\\omega}(s,\\mu\\_{\\theta}(s))$$ \nUpdate target networks\n \n*ϕ*targ ← *ρ**ϕ*targ + (1 − *ρ*)*ϕ*updated \n \n*θ*targ ← *ρ**θ*targ + (1 − *ρ*)*θ*updated \n \n*ω*targ ← *ρ**ω*targ + (1 − *ρ*)*ω*updated \nDeep Trajectory-Based Stochastic Optimal Control\n================================================\nDeep Trajectory-Based Stochastic Optimal Control (DTSOC), proposed in (also see for an exposition), is a method for solving stochastic control problems through formulating the control problem as optimizing over a computational graph, with the sought controls represented as (deep) neural networks. The approximation power of deep neural networks can mitigate the curse of dimensionality for solving dynamic programming problems.\nWe briefly review the setup of the method here. Consider a stochastic control problem given by the following underlying stochastic dynamics,\n \n*s**t* + 1 = *f*(*s**t*, *a**t*, *ξ**t*) \nwhere, *s**t* is the state, *a**t* is the control (agent’s action) and *ξ**t* is a stochastic disturbance impacting the period between time index *t* and *t* + 1.\nIn the models for the hedging instruments derived from (discretized) SDEs, the *ξ**t* will be the Brownian increments in the (discretized) SDEs for the time step from index *t* to *t* + 1, Δ*W**t* as a discretization of *d**W**t*. The price of the hedging instrument as well as the amount of them held would be part of the state and the action would either directly give the new amount to be held or an increment or rate that would allow the new amount to be computed.\nWe assume that the actions are given as deterministic or stochastic feedback controls *a**t* = *π**t*(*s**t*∣*θ**t*) or *a**t* ∼ *π**t*( ⋅ ∣*s**t*, *θ**t*). One can extend the state *s**t* with path-dependent extra state that can be computed from current and previous state, action, and disturbances; and also with particular precomputed features that might lead to more efficient training of agents or more efficient agents which also extends the set of controls that can be written as feedback controls.\nThe actions can be constrained to come from a set of admissible functions:\n \n*a**t* ∈ A*t* = {*a**t* : *g*(*s**t*, *a**t*) = 0, *h*(*s**t*, *a**t*) ≥ 0}, \nwhere *h*(*s**t*, *a**t*) and *g*(*s**t*, *a**t*) are inequality and equality constraints. We assume that these constraints are already taken into account in the feedback controls such that *π**t* will be an admissible action or give a distribution over admissible actions.\nWe assume step-wise contributions given by *c**t* = *c**t*(*s**t*, *a**t*, *s**t* + 1) and also a final contribution *c**T*(*s**T*).\nGiven a deterministic or probabilistic policy in feedback form that gives admissible actions, we generate an episode \n*s*0, *a*0, *s*1, *a*2, ⋯, *a**T* − 1, *s**T* \n and obtain a total contribution\n \n*C* = ∑*t* = 0*T* − 1*c**t*(*s**t*, *a**t*, *s**t* + 1) + *c**T*(*s**T*). \nThe stochastic optimal control problem now minimizes (or maximizes) a loss function *l* of the expected total contribution, conditional on starting state *s*0. If *s*0 is not fixed, this will be a function of *s*0. We thus try to minimize $\\E[l(C)|s\\_0=s]$ or $\\E[l(C)]$ varying the policies *π**t*. For quadratic hedging, the loss function is *l*(*x*) = *x*2. Other loss functions can be considered as long as they can be meaningfully optimized over by appropriate mini-batch approaches.\nWith some given functional form (such as DNN) with an appropriate parametrization (for example, determine weights and biases while activation functions are fixed for complete feed-forward DNN) as deterministic policy, we obtain\n \n$$\\E\\left[l(C)\\right] = \\E\\left[ l\\left(\\sum\\_{t=0}^{T-1} c\\_t\\left(s\\_t,\\pi\\_t(s\\_t|\\theta\\_t),s\\_{t+1}\\right) + c\\_T(s\\_T)\\right) \\right] =: {\\cal L}\\left(\\left\\{ \\theta\\_t\\right\\}\\_{t=0}^{T-1} \\right) =: {\\cal L}(\\Theta) \\label{DNN\\_loss}$$ \nOne now jointly optimizes over all policies {*π**t*}*t* = 0*T* − 1 respective over all parameters of such {*θ**t*}*t* = 0*T* − 1 to optimize the loss function as applied to the total contribution (if *s*0 is not fixed, this will also be a function of *s*0 and we would need to take an appropriate expectation over *s*0 or keep *s*0 as a parameter).\nThe controls at each time step could be stacked into a computational graph with a loss function given in. For each roll out of the control problem, this computational graph takes the sequence of disturbances {*ξ**t*}*t* = 0*T* − 1 as input and gives the loss function as applied to the total contribution inside the expectation in as output. Figure [DTSOCCG] shows the computational graph to compute the loss ${\\cal L}$ and has the following features:\n* The deterministic policy at time step *t* is represented by some network with appropriate architecture (shown in a pink box) *s**t* → *a**t* with parameters *θ**t* that are trainable (can be optimized over)\n* The transition of the system to a new state, (*s**t*, *a**t*) → *s**t* + 1 based on the system dynamics is encoded in the connections from *s**t*, *a**t*, and the random disturbance *ξ**t* (shown in blue) to *s**t* + 1.\n* *s**t* and *s**t* + 1 will be input to *c**t* as shown in the graph\n* Defining the cumulative contribution up to time *t* as,\n \n*C**t* = ∑*τ* = 0*t**c**τ*(*s**τ*, *a**τ*, *s**τ* + 1), \nthe horizontal connections on top of the network, (*s**t*, *a**t*, *C**t*) → *C**t* + 1 sums up the time *t* contribution and gives the total accumulated contribution *C**T* at the end of the episode (when *t* = *T*).\n* The total accumulated contribution *C**T* is then passed through a loss function (shown in light brown) and gives the loss $\\cal L$ (shown in gray) as final result.\nNote that based on a discretization {0 = *t*0 < *t*1, ⋯, *t**p* = *T*} of the time horizon, the computational graph will have *p* layers (with *p* embedded DNN) and *p* × ∑*t* = 0*T* − 1*N**t* trainable parameters. After the loss ${\\cal L}$ has been computed as in the above computational graph, standard deep learning frameworks such as TensorFlow or PyTorch can now use the computational graph to generate path-wise gradients with respect to all trainable parameters.\nPseudo-code for DTSOC is given in Algorithm [DSOCtraining] below.\n[H] [DSOCtraining]\n**Initialize:** Network weights {*θ**m*, *m* = 0, ⋯, *T* − 1} *n**b**a**t**c**h* = 0 *b**a**t**c**h**l**o**s**s* = 0 *m* = 0 *C*− 1 = 0 *a**m* = *π**θ**m*(*s**m*)*s**m* + 1 = *f*(*s**m*, *a**m*, *ξ**m*) with sampled noise *ξ**m*. *C**m* = *C**m* − 1 + *c**m*(*s**m*, *a**m*, *s**m* + 1) m++ *C**T* = *C**T* − 1 + *c**T*(*s**T*) *b**a**t**c**h**l**o**s**s* + = *l*(*C**T*) Calculate loss for batch \nLoss = *b**a**t**c**h**l**o**s**s*/(batch size) \n Calculate gradient of Loss with respect to *θ* Back propagate updates for {*θ**m*, *m* = 0, ⋯, *T* − 1} epoch++ **return** optimized weights {*θ**m*\\*, *m* = 0, ⋯, *T* − 1}\nThe contributions *c**t*(*s**t*, *a**t*, *s**t* + 1) correspond to the step-wise rewards in the reinforcement learning approaches. Just like there, only the sum counts in terms of optimal solutions.\nThe “deep hedging” paper presents a trajectory based empirical deep stochastic optimal control approach to minimizing some global objectives related to replication and/or risk management of some final payoff. It mentions a version with loss function in equation (3.3) and mentions some multi-dimensional quadratic hedging example in section 5.4, but does not present detailed discussion or tests for quadratic hedging.\nRelationship to FBSDE Formulation of Stochastic Control\n-------------------------------------------------------\nWe here repeat and adapt the discussion from. In general, one can consider a stochastic control problem in which some functional defined by running and final costs (which depend on the evolution of some controlled forward SDE and on the control) is optimized over that control. This leads to coupled *forward backward stochastic differential equations* (FBSDE) and non-linear PDEs (See or for an introductory treatment). In our setting, the control that the agent tries to optimize does not impact the forward SDEs describing the evolution of the prices of the instruments, it only impacts the trading strategy, leading to a controlled backward SDE only.\nWith *X**t* being the factors and prices for the (hedging) instruments and *Y**t* being the value of the hedging strategy, we have the system (see )[6](#fn6),\n \n*d**X**t* = *μ*(*t*, *X**t*)*d**t* + *σ*(*t*, *X**t*)*d**W**t* \n \n*d**Y**t* = − *f*(*t*, *X**t*, *Y**t*, Π*t*)*d**t* + Π*t**T**σ*(*t*, *X**t*)*d**W**t* \nwhere Π*t* plays the role of a control or strategy and the functional to be optimized (typically, minimized) is\n \n*J**F*(Π*t*, Πfinal, …) = *E*(∫0*T*rc(*s*, *X**s*, *Y**s*, Π*s*)*d**s* + fc(*X**T*, *Y**T*, Πfinal)). \nFor the example of an European option and quadratic hedging, *Y**T* has to replicate the appropriate payoff *g*(*X**T*) in the mean square sense, i.e. fc(*X**T*, *Y**T*, ⋅ ) = (*Y**T* − *g*(*X**T*))2. If the market is complete, *Y**T* will be perfectly replicable and thus the exact loss function or final cost does not matter as long as it is zero when *Y**T* is perfectly replicated.\nOne can define \n*J**t* = ∫0*t*rc(*s*, *X**s*, *Y**s*, Π*s*)*d**s* \n or \n*d**J**t* = rc(*t*, *X**t*, *Y**t*, Π*t*)*d**t* \n and add it to the stochastic system, looking for a minimum of \n$$\\E \\left( J\\_T + {\\mathsf{fc}}({X}\\_T,Y\\_T,\\Pi^{\\mathsf{final}})\\right).$$ \nOne can derive FBSDE characterizing the optimal controls (both primal and dual/adjoint) as well as PDEs characterizing them, but we will here concentrate on approaches that directly optimize over the given system for *X**t*, *Y**t*, Π*t*, and *J**t*.\nUpon time-discretization, one obtains stochastic control problems defined on (controlled) FBSΔE (Δ standing for “difference”) where now the running cost can depend on the forward and backward components and the control at both the beginning and end of each time-period.\nApplying a simple Euler-Maruyama discretization for both *X**t* and *Y**t*, we obtain\n \n*X**t**i* + 1 = *X**t**i* + *μ*(*t**i*, *X**t**i*)Δ*t**i* + *σ**T*(*t**i*, *X**t**i*)Δ*W**i* \n \n*Y**t**i* + 1 = *Y**t**i* − *f*(*t**i*, *X**t**i*, *Y**t**i*, Π*t**i*)Δ*t**i* + Π*t**i**T**σ**T*(*t**i*, *X**t**i*)Δ*W**i* \nThis can be used to time-step both *X**t* and *Y**t* forward.\nNow, quadratic hedging means that one minimizes the squared differences *E*(∣*Y**T* − *g*(*X**T*)∣2) and the form of this final loss function matters since one in general can no longer perfectly replicate *g*(*X**T*).\nSimilarly, the running costs need to be accumulated\n \n*J**t**i* + 1 = *J**t**i* + rc(*t**i*, *X**t**i*, *Y**t**i*, Π*t**i*)Δ*t**i* \nand the stochastic optimal control problem will try to minimize \n$$\\E \\left( J\\_T + {\\mathsf{fc}}({X}\\_T,Y\\_T,\\Pi^{\\mathsf{final}})\\right).$$ \nA time-discrete setting allows one to incorporate more general transaction costs for *Y**t* by more complicated generators *f*\n \n*Y**t**i* + 1 = *Y**t**i* − *f*Δ*t*(*t**i*, Δ*t**i*, *X**t**i*, *X**t**i* + 1, *Y**t**i*, *Y**t**i* + 1, Π*t**i*, Π*t**i* + 1) + Π*t**i**T**σ**T*(*t**i*, *X**t**i*)Δ*W**i* \nand also more complicated running costs\n \n*J**t**i* + 1 = *J**t**i* + rc(*t**i*, Δ*t**i*, *X**t**i*, *X**t**i* + 1, *Y**t**i*, *Y**t**i* + 1, Π*t**i*, Π*t**i* + 1)Δ*t**i*, \nwhich could include running costs that depend on the profit and loss of some strategy across the corresponding time interval.\nIn, path-wise deepBSDE methods for such problems are discussed, at least applied to pricing and risk management where there is only a final cost (or a cost at the earlier of reaching a barrier or maturity) - as in the quadratic hedging setup considered here.\nDeepBSDE methods represent the strategy Π*t* as a DNN depending on appropriate state *X**t* (or features computable from such state). Path-wise forward deepBSDE methods generate trajectories of *X* and *Y* starting from initial values *X*0 and *Y*0 according to the current strategy. They then use stochastic gradient descent type approaches such as ADAM to improve the strategy until an approximate optimum is reached. If the initial wealth *Y*0 is not given, it will be determined by the optimization as well. If the starting value *X*0 of the risk factor vector is fixed, *Y*0 would be a single value, otherwise it would be a function of *X*0. The optimization problem would typically represent this function as a DNN.\nDerived so far for certain kinds of final costs or where one attempts to replicate the final payoff as well as possible, path-wise backward deepBSDE methods make the same assumptions, but on each generated forward trajectory of *X*, they start with an appropriate final value of *Y**T* (*Y**T* = *g*(*X**T*) for the final payoff case), compute a corresponding trajectory of *Y**t* by stepping backward in time, and try to minimize the range of *Y*0. It appears that the strategies computed by the backward deepBSDE methods also perform well when used and tested with the quadratic hedging loss function in a forward stepping approach.\nQuadratic hedging for European options has been considered with forward and backward path-wise methods in for linear pricing and in for nonlinear pricing (differential rates) while forward path-wise methods were introduced earlier by. The barrier option case is treated with forward methods in.\nOne difference between the setup discussed in this section and other approaches considered in the paper is that here the backward SDE or SΔE is written in such a way that it uses parts of the forward model (i.e. *σ**T*(*t**i*, *X**t**i*)Δ*W**i*) and might not as written satisfy self-financing exactly but only up to discretization accuracy. In this way, this is a (more) model-based approach. However, one can rewrite the BSDE for *Y* so that it only uses the stochastic increment of *X* (i.e., written in *d**X* rather than using model details about *X*) and one can rewrite the BSΔE so that it only requires observations of *X* at trading times *t**i* and perfectly preserves self-financing similarly to what we wrote in earlier subsections, obtaining methods that will be more similar to the ones discussed there.\nExperimental Setup and Model Specification\n==========================================\nWe consider the example of a European call option with strike price *K* and maturity *T* on a non-dividend-paying stock. The strike price and option maturity are considered as fixed parameters. It is assumed that the risk-free rate is zero and that the option position is held until maturity. Rebalancing of the hedging portfolio is allowed at fixed (often regular) times and trades are subject to transaction cost proportional to the trade size. The trained hedging agent is expected to learn to hedge an option with this specific set of parameters. It is possible to train parametric agents that can hedge a parametrized set of options (such as calls with various strikes), but we will not do so here. We assume the Black-Scholes or SABR model for the simulation environment where the stock dynamics is given by\n \n$$dS\\_{t} = \\mu S\\_t \\; dt + \\sigma\\_t S\\_t \\; dW\\_t \\label{SDE:Stock},$$ \nwith either a constant *σ**t* = *σ* (Black-Scholes) or a stochastic volatility *σ**t* given by the SDE\n \n$$d\\sigma\\_t = \\frac{\\eta}{2} \\sigma\\_t dW^2\\_t$$ \nand trading (buying and selling) of stock incurs a transaction cost which is assumed to have the functional form,\n \ncost(*S**t*, *δ**H**t*) = *α*∣*S**t**δ**H**t*∣, \nwhere *δ**H**t* is the change in the stock position.\nThe default parameters for the stock, the option and transaction costs are as given in Table [BS-defaultparameters] with the additional parameters for the SABR model as in Table [SABR-defaultparameters].\n[htb]\n| l || l | Parameter & Value \n*μ* & 5 % (rate of return) \n*σ* & 20 % (Black-Scholes volatility) \n*i**r* & 0.0 % (interest rate) \n*S*0 & 100 \n*K* & 100 \n*T* & 30 (option maturity- days) \n*α* & 0.001 (transaction cost parameter) \n[htb]\n| l || l | Parameter & Value \n*σ*0 & 20 % (initial volatility) \n*η* & 0.95 % (volatility of volatility) \n*ρ* & 0.5 % (correlation of stock and volatility Brownian) \nState Space Selection and Model Architecture\n--------------------------------------------\nThere are at least two parts of state - one part describes the state of the environment and its evolution; the other part is provided as input to the training and trained agent; and these two parts certainly overlap. Certain features or transformations of features can be added that might enabled faster training or more optimal agents. One can also train agents that are robust against certain changes in parameters, but we will not do so here.\nThe state of the environment and the action are characterized by the following variables, given at each time step:\n* Time *t*, 0 ≤ *t* ≤ *T*,\n* Stock price at time *t*,\n* Current stock holding[7](#fn7) *H**t*.\nOne can parametrize the trading strategy directly by specifying *H**t*, the quantity of stock held at time *t* or by assuming that amount rebalanced is proportional to time passed and parametrizing the rebalancing rate. The quantity of stock held should be Markovian and thus directly parametrizing it might have certain advantages. The two parametrizations are related through the equation,\n \n$$\\label{rate\\_to\\_ratio}\nH\\_{t\\_{m+1}}^{\\theta\\_{m+1}}=H\\_{t\\_{m}}^{\\theta\\_{m}}+ \\dot{H}\\_{t\\_m}^{\\theta\\_{m}}\\Delta\\_t.$$ \nWe choose to parametrize the quantity of stock held. This is similar to how other implementations in the literature have used relatively shallow deep networks with three layers for parametrizing each individual control (see ). RELU was used as the nonlinear activation function in all hidden layers.\nDeep-QH agent is a feed-forward neural network with three hidden layers with 10, 15 and 10 neurons, with batch-normalization in each layer. The final output transformation is chosen as linear.\nFor RL-QH, the critic networks to learn the *Q* and *K* function and the actor network all have two hidden layers with 32 and 64 neurons, with batch-normalization before and after each layer. The *Q* and *K* function networks have linear output transformation while the actor/policy network has a sigmoid output transformation.\nModel Training\n--------------\nThe model was trained on a computer with 8 core i7- 11850 CPU $@$ 2.50GHz processor and 32.0 GB memory.\nFor the RL-QH model, both actor and critic are trained with respect to a mean squared error (MSE) loss with the TensorFlow implementation of ADAM optimizer, actor learning rate and critic learning rate are set to 1e − 4 and 1e − 4 respectively. The smoothing parameter for updating the weights of the target network is 1e − 5. The RL-QH model was trained over 50,000 episodes. The above parameters were kept fixed for both zero and non-zero transaction cost regimes.\nThe deep-QH model is trained using ADAM optimization with Pytorch off-the-shelf parameters. Batch normalization and dropout (*p* = 0.25) were applied to each layer. The learning rate was fixed at 1e − 3 initially and was decreased dynamically as training epochs progressed. The model was trained for 50,000 episodes.\nResults for Black-Scholes\n=========================\nWe trained the Deep-QH and RL-QH agents as described above and compared them against the Delta hedge agent (which is variance optimal in the zero transaction cost case), for an option with 30 days to maturity. We show histograms for the final mismatch between the final value of the trading strategy versus the required final payoff in figure [benchmarkRLvsdeepQHnoTC]. Negative values (to the left) mean that the strategy was worth more than the payoff and leading to a profit after paying out the required payoff. Positive values (to the right) mean that the strategy was worth less than the payoff and would lead to a loss after paying out the required payoff. We observe that all three agents have very similar average profit or loss, with Delta hedge being a bit more concentrated at zero profit or loss, with the Deep-QH spread out somewhat more. RL-QH sometimes leads to bigger profits but also allows larger losses, controlling P&L not as tightly.\nLonger Maturity\n---------------\nHere, we trained the agents to hedge options with shorter maturity (10 days) or longer maturities (60 days and 90 days) and show the results in figure [benchmarkRLvsdeepQHincreasingmaturity]. For shorter maturities, RL-QH outperforms on average while giving very similar results to 30 days for longer maturities. RL-QH consistently shows a far wider spread, showing that it does not control tails as well as the Deep-QH agent or the Delta hedge agent. All hedging agents allow a larger spread or variance for longer option maturities and as before, the Deep-QH agent allows a somewhat wider spread but still substantially more controlled than the RL-QH agent.\nIncreasing Volatility\n---------------------\nHere, we compared settings with increasing volatility of the underlying stock while keeping option maturity at 30 days, showing the results in figure [benchmarkRLvsdeepQHincreasingvolatility]. With increasing volatility, the RL-QH agent does worse on average, realizing fewer larger profits while still allowing larger losses. The variance for all agents increases with increasing volatility. The relative performance and shape relationship between Deep-QH and Delta hedge remains the same.\n[benchmarkRLvsdeepQHincreasingvolatility]\nIncreasing Transaction cost\n---------------------------\nFigure [benchmarkRLvsdeepQHincreasingTC] shows the performance of the RL-QH, Deep-QH and Delta hedge agents with increasing transaction costs. One can observe that both RL-QH and Deep-QH systematically outperform the Delta hedge agent, with the Deep-QH agent performing increasingly better against the RL-QH agent. While both RL-QH and Deep-QH have a peak corresponding to at least moderate gains and a substantial percentage of trajectories that end in gains, there is a heavier tail of larger losses, compare to the tails of the Delta hedge agent, for large enough transaction costs. While the behavior of the Deep-QH seems to be relatively smooth, RL-QH allows a set of more uncontrolled losses.\n[benchmarkRLvsdeepQHincreasingTC]\nResults for SABR\n================\nHere, the RL-QH and Deep-QH agents were trained on trajectories generated by the log-normal SABR model, where the stochastic volatility is not observed by the agents. We also implemented an agent that implements Bartlett’s Delta hedge with the approximation for implied volatility as described in an earlier section. The results are shown in figure [benchmarkRLvsdeepQHSABR]. One can observe that after training, RL-QH and Deep-QH agents on average perform as well as the approximately variance optimal Barlett’s Delta hedge agent. Both RL-QH and Deep-QH allow larger variance than the Barlett’s Delta hedge agent. Given that stochastic volatility is unobserved by the agents, at least part of this variance could be explained by that. We intend to closer investigate this in future work.\n[benchmarkRLvsdeepQHSABR]\nModel Robustness and Generalization\n-----------------------------------\nAs a simple model robustness test that tests whether agents trained on simpler models perform sensibly on more complicated models, we trained RL-QH and deep-QH agents on Black-Scholes models corresponding to the initial volatility in the log-normal SABR model and tested them against the variance-optimal Bartlett’s Delta hedge agent when trajectories are generated by the log-normal SABR model. Histograms[8](#fn8) for the final profit or loss are shown in figure [benchmarkRLvsdeepQHrobustness]. One can observe that all three agents lead to very similar average performance. Deep-QH agent’s performance is spread out more than the Barlett’s Delta performance, but still tightly peaked close to zero loss or gain. While RL-QH has similar average performance, its performance shows a very wide variance. It would be interesting to investigate this further to see how this performance comparison depends on parameters and also whether training on Black-Scholes models with varying parameters will lead to improved performance on the SABR test case.\n[benchmarkRLvsdeepQHrobustness]\nConclusion and Future Work\n==========================\nWe implemented and studied deep trajectory-based stochastic optimal control and deep reinforcement approaches to minimize the variance of the final hedging P&L. In particular, we implemented and applied an extension of the DDPG Actor-Critic approach to the case where the expectation of the second moment of the return is optimized rather than the expected return. We trained and tested on the Black-Scholes model and the log-normal SABR model. Without transaction costs, variance optimal strategies are known - for Black-Scholes, Delta hedging is optimal, while Bartlett’s Delta hedging with the exact implied volatility is variance optimal in the log-normal SABR model when only the underlier is used for hedging. With a good approximation for the implied volatility as available for the SABR model, Barlett’s Delta hedging with that approximation is very close to variance optimal. Deep-QH and RL-QH agents match the (approximate) variance optimal strategies in average cost with comparable (deep-QH) or wider (RL-QH) variance and range, with RL-QH allowing a wider range of both extreme gains and losses. For non-zero and increasing transaction costs, both RL-QH and deep-QH outperform the variance optimal Delta hedging on average, with deep-QH doing so more consistently than RL-QH in the Black-Scholes case. Similar results are seen for log-normal SABR which is and example for an incomplete market (if only hedged with stock). We finally tested agents trained on Black-Scholes model with the initial volatility from the SABR model but tested them on SABR trajectories. On average, RL-QH and deep-QH still match the performance of Bartlett’s Delta hedge, however showing larger variance (for deep-QH) and dramatically larger variance (RL-QH).\nThis paper and these results suggest areas of additional work. A more complete study of the SABR model and other more complicated and incomplete models, studying further behavior with increasing transaction costs and with varying maturity, strike, and volatility would be in order. It would be interesting to see whether other architectures, algorithmic choices in the RL algorithm, and hyper-parameters and choices in the training would allow better control of variance and outliers for the RL approaches and improve the deep-QH results even further. A second set of questions concerns latent factors such as the stochastic volatility in SABR. What would the impact be if the stochastic volatility would be treated as observable and provided to the agents? What would the impact be of adding additional hedge instruments such as a variance swap or one or several European options to complete the market? Would a more robust training (over varying volatility parameters, for instance) under the Black-Scholes model lead to better performing agents under the SABR model with better variance control? Would adding volatility estimates that are computed from the observed or generated sequence of spot prices as input features to the agent allow that training of agents that are more robust against model choice and specification?\nAlso, a set-up that allows easier exploration of models, trading strategies, and objective functions, such as an extension of a generic simulation framework as used in would make such studies easier.\nFinally, how would a transaction-cost-aware delta strategy (as the Leland model and strategy for Black-Scholes) perform compared to RL and deep trajectory-based stochastic optimal control approaches? Would training of RL and deep-QH approaches on different transaction costs and giving transaction cost parameters as input to those agents lead to improved training and behavior in the presence of transaction costs?\nAcknowledgments\n===============\nThe author acknowledges and appreciates the contributions by Ali Fathi (who during these contributions was working at Wells Fargo) who implemented the reinforcement learning and deep trajectory-based stochastic optimal control algorithms, the Black-Scholes and log-normal SABR settings, and generated the results shown in this paper. The author appreciates the assistance of Abdolghani Ebhrahimi of Wells Fargo with the specification, adaptation, and running of the reinforcement learning algorithms for the second moment of the MDP return.\n---\n1. All contents and opinions expressed in this document are solely those of the author(s) and do not represent the view of Wells Fargo Bank NA.[↩](#fnref1)\n2. One can also add other features and information to the state space that the agent could potentially use to make better decisions or that allow the agent to be trained more easily. Also, instead of *S**t**i*, one can use *S̃**t**i* as defined below, which is *S**t**i* discounted back to time 0.[↩](#fnref2)\n3. See also for a discussion of possible approaches together with associated model risks.[↩](#fnref3)\n4. A call payoff discounted back to time zero can be written as *D**T*(*S**T* − *K*)+ = (*S̃**T* − *K̃*)+ which can be understood as a call on *S̃**T* with discounted strike *K̃* = *D**T**K*, needing no discounting of the payoff. The Black-Scholes formula then simplifies to *c*BS, *t*(*S̃**t*, Σ*t*; *T*, *K̃*) = *S̃**t*Φ(*d**t*+(*S̃**t*, Σ*t*)) − *K̃*Φ(*d**t*−(*S̃**t*, Σ*t*)). Since *K̃* and *T* are fixed once we fix the call option to be hedged, we omit these two arguments to our function since they are already bound.[↩](#fnref4)\n5. Instead of optimizing a function of the sum, one can also optimize over some characteristics of the distribution of the sum, such as a CVaR.[↩](#fnref5)\n6. In this subsection, we use notation from the FBSDE literature as adapted to the pricing and hedging domain and do not follow the generic notation for RL or trajectory-based approaches. The state *s**t* in RL or trajectory-based approaches would contain *X**t*, *Y**t*, Π*t*, *J**t*, and whatever is needed to compute terms and costs (or equivalent information), the action/control would be some parametrization of Π*t*, the stochastic disturbance *ξ**t* would be the *d**W**t* or Δ*W**i*.[↩](#fnref6)\n7. Given by or impacted by agent’s action in previous step(s).[↩](#fnref7)\n8. Figure [benchmarkRLvsdeepQHSABR] shows results for a larger number of option contracts while figure [benchmarkRLvsdeepQHrobustness] shows results for one option contract, thus the ranges and variances of the RL-QH and deep-QH agents trained on Black-Scholes are substantially larger than for the agents trained on SABR as in figure [benchmarkRLvsdeepQHSABR] - the range and variance of the Bartlett’s Delta hedge agent in both figures are the same.[↩](#fnref8)\n\nReinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging\n======================================================================================\n=1\nWe consider two data driven approaches, Reinforcement Learning (RL) and Deep Trajectory-based Stochastic Optimal Control (DTSOC) for hedging a European call option without and with transaction cost according to a quadratic hedging P&L objective at maturity (“variance-optimal hedging” or “final quadratic hedging”). We study the performance of the two approaches under various market environments (modeled via the Black-Scholes and/or the log-normal SABR model) to understand their advantages and limitations. Without transaction costs and in the Black-Scholes model, both approaches match the performance of the variance-optimal Delta hedge. In the log-normal SABR model without transaction costs, they match the performance of the variance-optimal Barlett’s Delta hedge. Agents trained on Black-Scholes trajectories with matching initial volatility but used on SABR trajectories match the performance of Bartlett’s Delta hedge in average cost, but show substantially wider variance. To apply RL approaches to these problems, P&L at maturity is written as sum of step-wise contributions and variants of RL algorithms are implemented and used that minimize expectation of second moments of such sums.\nIntroduction\n============\nData-driven hedging has attracted wide and deep interest as an application of reinforcement learning (RL) and similar approaches to derivatives pricing and risk management, see for instance,, or. These approaches have advantages, such as (1) that they can work with observed, simulated, or generated data for the behavior of the instruments used for hedging and risk management, rather than proceed in model-specific fashion, (2) that the value and other characteristics of a trading strategy over time can be easily modeled given such data, even for realistic models of incomplete markets where trades cause friction and attract transaction costs, and (3) that hedging or risk management measures of such strategies can be easily computed and used as objectives to be optimized over or controlled for. These measures could quantify how the strategies perform as measured against book-keeping or market prices of the to-be-hedged instruments or quantify the profits and losses when following that strategy to hedge or risk-manage the to-be-made or to-be-received payoffs for the to-be-hedged portfolios. In the later case, the behavior of such strategies can be modeled and optimized over even as consistent prices or sensitivities might be difficult or impossible to compute in conventional models with conventional methods. This allows the training of data-driven agents that optimize said hedging and risk management measures and that perform as well as or better than conventional sensitivy-based and model-based hedging approaches. Data-driven agents can also take other factors into account and automate or support hedging in situations where conventional hedging approaches require ad-hoc solutions and tweaks by traders and risk managers.\nSeveral banks and other players in the financial industry have started to investigate and apply these new approaches to hedging and risk management, such as reported in. However, the resulting agents, their behavior, and associated risks of use, including model risk, need to be studied and understood, so that appropriately designed and understood data-driven hedging and risk-management solutions can be applied safely and successfully. Here, we will study how RL and Deep Trajectory-based Stochastic Optimal Control (DTSOC) approaches can be applied to hedging problems, in particular hedging to control the quadratic deviation (variance) of the final profit-and-loss after having made and received all payoffs of the instrument or portfolio to be hedged (“quadratic final hedging”).\nScope of This Work\n------------------\nThis paper concentrates on hedging strategies that control the quadratic deviation (variance) of the final profit-and-loss after having made and received all payoffs of the instrument or portfolio to be hedged (“quadratic final hedging”). The setting will apply to other trading strategies with other objectives also.\nTo define the setting, we will model how the prices of the hedging instruments evolve, how the trading strategy to do so performs (allowing transaction costs), and define the final objective to be optimized over in terms of the prices, costs, etc. involved in the strategy.\nTo set up training of agents for final quadratic hedging, one only needs to model the prices of the hedging instruments, the trading strategy, and the payoffs of the to-be-hedged instrument or portfolio. This is different from the setting considered in, where one also needs to model the prices of the to-be-hedged instruments either with a consistent model or a book-keeping model. In situations where the to-be-hedged instrument and closely related instruments are not liquidly quoted nor traded (either on exchange or on some OTC system) and where no appropriate and flexible enough book-keeping models capturing the important features of the instruments and the trading (such as transaction costs) are available, final quadratic hedging is applicable (or anything that is defined purely in terms of a strategy involving the modeled prices of the hedging instruments) while step-wise mean-variance hedging would not be applicable.\nHere, we assume we have a model of how the appropriate risk factors and prices for the hedging instruments evolve as needed to compute the hedging objectives. This means that we have some implementation that can generate as many trajectories of these instrument prices and other risk factors as needed. Based on this data, one can simulate the trading strategy and determine its performance and thus the objective we will optimize over. It is assumed that the trading from the strategies does not impact how the risk factors and the prices of the hedging instruments evolve but only how the trading strategy and its results evolve, as is commonly assumed. We only assume that we can obtain trajectories as needed and do not need any information about how the model is obtained, setup, or run. Thus, either a well-specified traditional model could be used or some generative model trained on appropriate data as in or. We will work here with trajectories simulated from a Black-Scholes model or a log-normal SABR model while leaving synthetic data from generative models to future work.\nWe will test and validate our agents on simulated data from such models. Whether agents trained on simulated or generated data can perform well on observed data is an interesting research question and is left to future work. There is some work that shows that at least in some circumstances agents trained on simulated or synthetic data can perform well when run against observed market data.\nThe agents themselves are operating in a model-free, data-driven setting, since they only are provided generated trajectories and rewards rather than details about the model. Given our setting, we are in a rich data regime (even unlimited data regime), corresponding to a well defined conventional or generative model that is used to generated as much data as wanted.\nRelated Work\n------------\nWe discuss here some of the existing work in the literature on the use of Reinforcement Learning and Deep Trajectory Based Stochastic Optimal control to pricing and hedging of options. For some broader survey of RL applications in finance, see.\nIn, a European call option in the Black-Scholes model (trading at discrete times, no transaction costs) is hedged and priced using a *Q*-learning method (there, called QLBS method). The objective is minimizing the sum of the initial cash position and the weighted discounted sum of variances of the hedging portfolio at all subsequent time steps. This setting and objective can be identified with a corresponding MDP setup and thus solved with standard RL/DP approaches such as *Q*-learning. The price at maturity has to be equal to the final payoff and prices respective *Q*-functions at earlier times are determined by Dynamic Programming or other standard RL approaches.\n defines the objective as a combination of the expected final P&L and the expected variance of the final P&L. They use versions of *Q*-learning and other methods for the expected sum of contributions and the expected square of sums of contributions to address this problem. We will use similar approaches to optimize the variance of the final P&L of the hedged portfolio, but did not discuss or apply their approaches to final quadratic hedging. test their methods on data simulated from the Black-Scholes and log-normal SABR model and compare their results to Delta hedging with current implied volatility and Bartlett’s Delta hedging with current implied volatility.\nMany works consider neural networks as a numerical tool to accelerate calculation tasks and to solve generic classes of problems, such as model calibration, solving PDEs, solving stochastic optimal control problems more efficiently and in higher dimensions. Among these, we distinguish the line of works initiated in where neural networks are used to solve certain stochastic optimal control problems numerically. More concretely, the DTSOC method in is a setup for trajectory based empirical deep stochastic optimal control with both step-wise and final objectives/costs. Finally, the “deep hedging” paper presents a trajectory based empirical deep stochastic optimal control approach to minimizing some global objectives related to replication and/or risk management of some final payoff, according to risk measures such as CVaR/expected shortfall, expected utility, etc. While quadratic hedging is mentioned, the method(s) are not applied to any variance-optimization/quadratic hedging problem in that paper.\nFramework\n=========\nGeneral Setup\n-------------\nWe consider here dynamic hedging, trading, and/or risk management problems defined in terms of some instrument or portfolio to be hedged or otherwise risk managed. In our examples, we assume that the instruments are European options; but one could handle portfolios or such or more path-dependent instruments in a very similar fashion. In these problems, there is an universe of given instruments (including cash/bank accounts that allow also negative balances) and the agent has to decide how much of each instrument to hold at each time to achieve a certain hedging, trading, or risk management objective. We assume, as is commonly done, that there is a given fixed set of times at which trades can occur and we allow that such trades might incur transaction costs. We also assume that the strategies are self-financing. The objectives under consideration can be defined from contributions at one or several fixed time horizons or each trading period and/or trading time might have a contribution associated to it. Here, we will consider objectives that are associated with the one final payment time of the to-be-hedged instrument. While the objective can be a simple mean-square hedging error as here, it could also be defined based on a risk measure (such as CVaR) or some moments or percentiles of an appropriately defined hedging outcome.\nThis is a stochastic sequential decision problem. The prices of the hedging instruments (here, stocks) will follow some stochastic process with some given model or generator. These prices can be observed and other appropriate risk factors could be computed based on the history of such prices. Now, given these observed prices and computed risk factors, the agent needs to decide in time for each trading time how much of each hedging instrument to buy and/or sell; or, equivalently, how much of each hedging instrument should be held immediately after that trading time; taking into account transaction costs and other trade impacts. (Thus, holding sizes or trade sizes would be appropriate decision variables.) Given to-be-achieved holding size or chosen trade size and the cash at hand before the trade, the cash remaining after (or being borrowed) can be computed by self-financing and funding strategy, the value of the holdings and of the cash at the end of the no-trade interval follows from the stochastic evolution of the prices of the hedging instruments and the interest on the cash or loan account, up to final maturity.\nTrading Strategy for the Hedging Portfolio\n------------------------------------------\nWe consider the case of a hedging portfolio. This means that we are given some description of the to-be-hedged instrument or portfolio. This description in general consists of random variates (*P**t*)*t* ∈ T describing the contractually-agreed payoffs under that portfolio, where T is the set of their payment times. We assume here that we are treating European options: there is only one payoff at some maturity *T* which we denote *P**T*.\nAs in, it is assumed that we are given “hedging” instruments (say, one or several stocks), we denote their prices by *S**t* (in general a vector), and denote the holding of them in our strategy at time *t* with *H**t**S* (a vector of corresponding length). For our examples, we will typically assume a single hedging instrument. Finally, we assume that excess cash from the trading strategy is deposited in a money-market account with a certain given interest accrual and discount factor (and that negative cash balances can be borrowed at the same interest accrual and discount factor). We assume that we initially start with some initial cash *Y*0 but no holdings in the hedging instruments (i.e., *H*0−*S* = 0). The trading times are given as *t**i* with *t*0 = 0. If referred to, *t*− 1 is a time just before time 0 with no holding in hedging instruments.\nThen, at each time *t**i* when one needs to decide what to trade, one can observe the prices *S**t**i* of the hedging instruments. At the same time the holdings in these instruments are still given by *H**t**i* − 1*S*. Now one has to decide how much (*H**t**i**S*) shall be held for the next no-trading period. Here, we assume that these trades are made instantaneously at *t**i*+. The minimal state *s**i* for this decision has to include at least (*t**i*, *S**t**i*, *H**t**i* − 1*S*).[2](#fn2) The action has to either describe *H**t**i**S* or how to obtain *H**t**i**S* from *H**t**i* − 1*S*.\nConsider time *t**i* + 1, right before the trade: The value of the hedging part of the strategy then is \n \n$$Y\\_{t\\_{i+1}} = \\frac{\\left( Y\\_{t\\_{i}} - H^S\\_{t\\_i} S\\_{t\\_i} - \\mathsf{TC}(H^S\\_{t\\_i},H^S\\_{t\\_{i-1}},S\\_{t\\_i}) \\right)} {\\mathsf{DF}\\_{i,i+1}} + H^S\\_{t\\_i} S\\_{t\\_{i+1}}$$ \nThis reflects that at the previous trade time *t**i*, the portfolio was rebalanced to the appropriate holdings *H**t**i**S*, transaction costs were charged TC(*H**t**i**S*, *H**t**i* − 1*S*, *S**t**i*), the thus resulting cash balance *Y**t**i* − *H**t**i**S**S**t**i* − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S**t**i*) attracts step-wise interest (interest accrual corresponds to dividing by appropriate step-wise discount factor *D**F**i*, *i* + 1). At the same time, the holdings in the hedging instruments are now valued based on the new price of the hedging instruments.\nIt helps to rewrite the equation as follows as in :\n \n$$Y\\_{t\\_{i+1}} = \\frac {1} {\\mathsf{DF}\\_{i,i+1}} Y\\_{t\\_i} + H^S\\_{t\\_i} \\left( S\\_{t\\_{i+1}} - \\frac {S\\_{t\\_i}} {\\mathsf{DF}\\_{i,i+1}} \\right)\n- \\frac{\\mathsf{TC}(H^S\\_{t\\_i},H^S\\_{t\\_{i-1}},S\\_{t\\_i})} {\\mathsf{DF}\\_{i,i+1}}$$ \nIn this completely linear setting without stochastic interest rates (and assuming that the transaction cost is linear in the instrument prices for positive multipliers *c*, TC( ⋅ , ⋅ , *c**S*) = *c*TC( ⋅ , ⋅ , *S*)), it is possible to “hide” the discounting in the value definitions in the following sense: with *Ỹ**t**i* = DF0, *i**Y**t**i* and *S̃**t**i* = DF0, *i**S**t**i*, we have\n \n*Ỹ**t**i* + 1 = *Ỹ**t**i* + *H**t**i**S*(*S̃**t**i* + 1 − *S̃**t**i*) − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S̃**t**i*) \nIn particular, this means for the increment\n \n$$\\tilde{dY}\\_{t\\_{i+1}} = H^S\\_{t\\_i} \\left( \\tilde{S}\\_{t\\_{i+1}} - \\tilde{S}\\_{t\\_i} \\right)\n- \\mathsf{TC}(H^S\\_{t\\_i},H^S\\_{t\\_{i-1}},\\tilde{S}\\_{t\\_i})$$ \nand we thus do not get any contribution or impact of the initial cash position *Y*0 beyond an additive shift.\nFinal Quadratic Hedging Objective\n---------------------------------\nWe will also discount the final payoff and denote it *P̃**T*. The discounted final hedging error is now:\n \n*L̃**T* = *P̃**T* − *Ỹ**T* \nand we look to minimize the mean square hedging error MSHE (“quadratic hedging”):\n \nMSHE(*Y*0, *H*⋅*S*) = E[*L̃**T*2] \nwhere the expectation is under the given model respective generator for the underlying hedging instruments. This hedging is also referred to as variance optimal hedging (where the variance is of the final hedging error). We typically avoid this name except when referring to the literature since it does not further specify which variance and use “final quadratic hedging” since this makes it clearer that we optimize over the expectation of the square of the final hedging mismatch.\nExtensions to stochastic interest rates and discounting, differential rates, more involved funding policies are possible, require further notation and details, and can be handled similarly, but are not necessary for the settings we discuss here. Differential rates and more involved funding policies could be included in the framework by adding bank accounts for positive and negative balances and other funding instruments to the hedging instruments and adding constraints to the decision variables (only positive holdings of positive bank account, only negative holdings of negative bank account i.e. bank debits or loans, restricting secured loans/repos to the amount held in the corresponding collateral, etc.).\nUnder some circumstances, the strategy might spend a lot of transaction costs to decrease the final hedging error (and thus requiring larger initial wealth *Y*0). In such cases, one might constrain *Y*0 or also minimize over initial wealth, or one might constrain total transaction cost - or also minimize over accumulated total transaction costs.\nSince the impact of initial wealth is only additive and separated, we can compute the optimal *Y*0 for each hedging strategy *H*⋅*S*. The formulation also covers the case in which *Y*0 has been chosen from the outside (prices set by other mechanism or the instrument is on the run and we start with the existing cash holdings (and/or stock holdings) assigned to the hedging of our portfolio) where we only optimize over the strategy for the remaining time until maturity.\nModeling the Hedging Instruments\n--------------------------------\nThe formulation of the objective function in this paper only depends on the final hedging error - i.e., the difference between the contractually agreed (discounted) payoff *P̃**T* and the final discounted value of the hedging strategy *Ỹ**T*. *Ỹ**T* in turn depends on the discounted prices of the hedging instruments and transaction costs earlier, but does not require any further information about the models for these instruments and prices.\nThe agent only needs to determine how many units to hold in each of the hedging instruments after rebalancing at each rebalancing time *t**i*, i.e. *H**t**i**S*. Alternatively, one could specify the units to be sold or bought at any given time together with the initial position; or assume that the amount sold or bought is proportional to the time between trades or some other variable and then specify that proportional trading rate in that setting. The minimal state for modeling this decision problem are these discounted prices (hedging instruments), the amount of the hedging instruments, time, and whatever state the models for these prices and the transaction costs need.\nAs for the models for the prices of the hedging instruments, one can model them under pricing or observational measure, with some differences in parameters and calibration or fitting. For the hedging instruments, one can use conventional quantitative finance models, possibly with hidden factors, such as Black-Scholes, Local Volatility Model, SABR model (with stochastic volatility), Heston model (with stochastic variance), quadratic rough Heston model, etc. One could use generative models such as GANs or appropriately trained neural Stochastic Differential Equations (SDEs) or similar. Finally, one could use historically observed data over one instrument or a cross section of similar instruments together with appropriate assumptions to generate possible future price movements or future prices.[3](#fn3)\nFor the to-be-hedged instrument, we only need to know the payoff at maturity and we do not need any model or method to compute the price of the to-be-hedged instrument at any other time, unlike in hedging and risk management setups that require book-keeping or reference prices for the to-be-hedged instruments. We consider at first simple examples such as the hedging of a short call option that was sold to some counterparty or the hedging of a long call option that was bought. Here, one only needs to specify the final payoff *P**T* = − (*S* − *K*)+ respectively *P**T* = (*S* − *K*)+. Black-Scholes models with constant or deterministic volatility and log-normal SABR model are considered here, but other models can be easily implemented.\nIn formulas, this would mean for the discounted price of the hedging instruments (“stocks”) in continuous time\n \n*d**S̃**t* = *σ**t**S̃**t**d**W**t* \nwith *σ**t* a constant or deterministic time-dependent function (Black-Scholes), or following the stochastic volatility SDE of a SABR model\n \n$$d\\sigma\\_t = \\frac{\\eta}{2} \\sigma\\_t dW^2\\_t$$ \nand in discrete time, appropriate log-Euler time-discretizations thereof (which are actually exact for *S̃**t* in this case). The Black-Scholes SDE also corresponds to the SDE for the underlier in a log-normal SABR model (with *β* = 1).\nIf one considers more complicated models based on conventional quantitative finance models, one would simulate the (discounted) underlying instruments with that model. If these models have additional factors, these factors would be added to the state. If some of these factors are latent or hidden factors, one would need to add some mechanism how these latent factors can be estimated or taken into account by some process on observed quantities, add these observed quantities to the state, and learn agents that only depend on observed and observable quantities, not the latent factors that will in general be unknown (and unknowable) to the agent. The stochastic volatility in our log-normal SABR test case can be considered either as a latent, unobserved state or as observed state. (We treat it here as a latent state that is not provided as input to the agent/policy.)\nIn general, trading in hedging instruments could impact the prices in those hedging instruments either temporarily or permanently. We assume here that the hedging instruments are traded liquidly and that the hedged instrument is such that it can be hedged without impacting the prices of the hedging instruments. To a certain extent, short-term price impact can be modeled by and absorbed into the transaction cost terms.\nAs discussed in abstract and introduction, here we focus on Black-Scholes and the log-normal SABR stochastic volatility models to investigate the agents and algorithms in a setting where the model and the features are simple enough, and will consider other models in future work.\nAnalytical Exact or Approximate Variance-Optimal Hedges\n=======================================================\nFor some models (Black-Scholes, SABR, Rough Bergomi) and some assumptions (time-continuous trading, accurate enough approximations), one can derive explicit analytical forms of exact or approximate variance-optimal hedges that minimize MSHE under those assumptions. derives the below formulas (for the case of zero interest rates) and applies them to the log-normal SABR and the Rough Bergomi model.\nAssume thus that we have given *C**t* as the time *t* price for an instrument with payoff *P**T* (which means in particular that *C**T* = *P**T*) and denote the discounted-to-time-zero price as *C̃**t* = *D**t**C**t*, with *D**t* being the time zero discount factor for time *t*, assuming also deterministic interest rates. Similarly denote the discounted-to-time-zero version of *S**t* by *S̃**t* = *D**t**C**t*. We assume that we simulate and operate under a pricing measure Q under which *C̃**t* and *S̃**t* are square integrable martingales. Given any dynamic continuous-time trading strategy *θ*⋅, the final discounted value of the hedging portfolio if started from initial wealth *w* would be\n \n*Ỹ**T* = *w* + ∫0*T**θ**u**d**S̃**u* \nThe final hedging error is thus\n \n*L̃**T* = *C̃**T* − *Ỹ**T*. \nWe minimize over the (risk-neutral) mean-square hedging error\n \nMSHE(*w*, *θ*⋅) = E[*L̃**T*2] \nand the minimizer under the above assumptions is called variance-optimal strategy *θ**V**O*. It is given as the Radon-Nikodym derivative of the finite-variation process ⟨*S̃*, *C̃*⟩*t* with respect to the finite-variation process ⟨*S̃*, *S̃*⟩*t*:\n \n$$\\theta^{VO}\\_t = \\frac {d\\langle\\tilde{S},\\tilde{C}\\rangle\\_t} {d\\langle\\tilde{S},\\tilde{S}\\rangle\\_t}.$$ \nFurther assume that *P**T* represents a call (similar derivations work for puts) and that we write its discounted price in terms of implied volatility Σ*t* by the Black-Scholes formula for zero interest rates (here denoted by *c**B**S*)[4](#fn4):\n \n*C̃**t* = *c*BS, *t*(*S̃**t*, Σ*t*; *T*, *K̃*) = *c*BS, *t*(*S̃**t*, Σ*t*). \nThis Σ*t* will be a stochastic process (and as a smooth function of martingales, will be a semi-martingale). We introduce\n \n$$\\begin{aligned}\n d\\_t^{\\pm} (\\tilde{S}\\_t,\\Sigma\\_t) &=&\\frac{\\log(\\tilde{S}\\_t/\\tilde{K})}{\\Sigma\\_t\\sqrt{T-t}}\\pm\\frac{\\Sigma\\_t\\sqrt{T-t}}{2}\\\\\n \\mathsf{Delta}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)&=&\\Phi(d\\_t^{+}(\\tilde{S}\\_t,\\Sigma\\_t))\\\\\n \\mathsf{Vega}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)&=&\\tilde{S}\\_t\\phi(d\\_t^{+}(\\tilde{S}\\_t,\\Sigma\\_t))\\sqrt{T-t}.\\end{aligned}$$ \nIn terms of Σ*t*, we obtain:\n \n$$\\label{QH\\_optimal\\_formula2}\n\\theta^{VO}\\_t= \\mathsf{Delta}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)\n+\\mathsf{Vega}\\_t(\\tilde{S}\\_t,\\Sigma\\_t)\\frac{d\\langle\\Sigma,\\tilde{S}\\rangle\\_t}{d\\langle\\tilde{S},\\tilde{S}\\rangle\\_t}$$ \nwith the corresponding Radon-Nikodym derivative in the second term.\nNote that if Σ*t* is constant or a deterministic function of time, the second term is zero and Delta*t* is a variance optimal strategy, as in Black-Scholes with constant or at most time-dependent volatility *σ*(*t*).\nThus, if we have an explicit Σ*t* and we can compute *d*⟨Σ, *S̃*⟩*t* and *d*⟨*S̃*, *S̃*⟩*t* explicitly, we have an explicit variance optimal strategy. We can also use close approximations $\\hat{\\Sigma}\\_t$ of Σ*t* and would expect the resulting *θ**t**A**V**O* to be a good approximation of *θ**t**V**O*.\nLog-normal SABR for the discounted-to-time-zero stock price *S̃**t* is written\n \n$$\\begin{aligned}\nd\\tilde{S}\\_t=&\\tilde{S}\\_t\\sigma\\_t dW\\_t\\\\\nd\\sigma\\_t=&\\frac{\\eta}{2}\\sigma\\_t dW^{2}\\_t,\\end{aligned}$$ \nwhere *d*⟨*W**t*, *W**t*2⟩*t* = *ρ**d**t* for *ρ* ∈ [ − 1, 1], we have\n \n$$\\begin{aligned}\n\\hat{\\Sigma}\\_t & = & \\sigma\\_t f(M\\_t) \\\\\nM\\_t &=& \\frac{\\eta}{\\sigma\\_t} \\log \\left( \\frac{\\tilde{K}}{\\tilde{S}\\_t} \\right)\\end{aligned}$$ \nwith *f* given in the SABR formula ( and ):\n \n$$f(y) = \\frac{y/2}{\\log\\left(1-\\rho\\right) - \\log\\left(\\sqrt{1+\\rho y+y^2/4} - \\rho - y/2\\right)}$$ \nThe (approximated) Radon-Nikodym derivative can be computed as follows :\n \n$$\\frac{d\\langle\\hat{\\Sigma},\\tilde{S}\\rangle\\_t}{d\\langle\\tilde{S},\\tilde{S}\\rangle\\_t} = \\frac{\\eta}{2\\tilde{S}\\_t} \\left( \\rho f(M\\_t) - (\\rho M\\_t+2)f'(M\\_t) \\right)$$ \nThus, we obtain an approximately variance optimal strategy:\n \n$$\\theta^{AVO}\\_t= \\mathsf{Delta}\\_t(\\tilde{S}\\_t,\\hat{\\Sigma}\\_t)+\\frac{\\eta}{2} \\phi(d\\_t^{+}(\\tilde{S}\\_t,\\hat{\\Sigma}\\_t))\\sqrt{T-t} \n\\left( \\rho f(M\\_t) - (\\rho M\\_t+2)f'(M\\_t) \\right)$$ \nThis turns out to be exactly Barlett’s delta in this case (*β* = 1). A similar approximation also exists for the one-factor Rough Bergomi model (which includes the Brownian Bergomi model as a special case).\nWhile our hedging problem is posed with discrete trading times and transaction costs, these exact or approximate explicit analytical strategies should be close to optimal for frequent trading without transaction costs in the parameter domains where the approximation performs very well. Otherwise, it should certainly serve as a useful reference point.\nDecision processes\n==================\nSettings\n--------\nMany problems such as the ones considered here can be cast in the context of an agent interacting with some environment based on some decision policy that tries to optimize (maximize or minimize, as the case may be) some function of a sum of step-wise, initial, and final contributions (“rewards”).[5](#fn5)\nTo specify abstractly such settings, one needs to specify the state of the environment, the set of possible inputs to the agent, and the set of possible actions by the agent. One also needs to specify how the environment transitions to a new state with new observations in reaction to an action (and this transition can be random and parts of the state might not be affected by the action) and what contribution each action and step will make to the sum of rewards (and that contribution might be stochastic as well).\nIn some situations, the state and the evolution of the environment might not be completely captured. This makes such a setting very hard to handle and standard results do not apply and we will not discuss such settings further. The transitions and rewards might potentially depend on all the states and actions that came before, presenting Nonmarkovian problems. Most algorithms and theory are presented for the Markovian case where both transitions and rewards only depend on the most recent previous state and the current action, and we will only discuss this setting here. In at least some cases it is possible to exactly or approximately capture path dependency by extending the state space (such as for up or down Barrier options keeping track of running maximum or minimum or whether the corresponding barrier has been breached yet or not) and then using Markovian methods. There are circumstances under which the agent only observes some part or function of the state etc., called partially observable decision processes. While such settings might be needed to treat settings in which there are latent parts of the state (such as an unobserved stochastic volatility), we will not cover their setting or theory here.\nOne sometimes introduces a reward discount factor *γ* ∈ (0, 1] if optimizing earlier rewards or costs is more important than optimizing later contributions, with *γ* = 1 corresponding to the undiscounted case. (One could potentially allow *γ* ≥ 1 to put more emphasis on later contributions.) Such *γ* also might help with the convergence or numerical behavior of the sum but a *γ* not equal to 1 does change the problem and its solution in general. In our setting, we are mostly interested in the *γ* = 1 case and might look at values close to 1 for numerical purposes.\nMarkov decision processes (MDPs) with deterministic rewards can be specified by the following:\n* *state space* S: describes the state of the evolution of the environment and what the agent takes as input to make a decision\n* *action space* A: describes the set of agent actions\n* *Transitions*:\n\t+ *Transition probability* P : S × A × S → [0, 1] or *transition density* *p* : S × A × S → R+: P(*s*, *a*, *s*ʹ) denotes the probability of the state transitioning from *s* to *s*ʹ as a result of taking the action *a*, and the next state *s*ʹ would be chosen based on the probability P(*s*, *a*, ⋅ ), or\n\t+ *Transition function* with explicit randomness: *s*ʹ = *f*(*s*, *a*, *ξ*) with *ξ* describing the randomness and stochastic impact that impacts the transition between *s* and *s*ʹ.\n* Deterministic *rewards* *r* : S × A × S → R: reward assigned to the transition (*s*, *a*, *s*ʹ).\n* *γ*: the *reward discount factor*.\nThe set of trajectories or realizations of MDP can be described as follows. One first starts with an initial state *s*0 that could be fixed or given as a random variate or otherwise varying. Then, starting with time index *t* = 0, *s**t* is given, agent chooses an action *a**t*, and environment generates a new state *s**t* + 1 = *f*(*s**t*, *a**t*, *ξ**t*) and a reward *R**t* + 1 = *r*(*s**t*, *a**t*, *s**t* + 1). Since there is randomness at least in the transition to the new state, *s**t* + 1 and *R**t* + 1 will be (at least partially) random and their distributions will depend on the actions taken. The agent’s decisions are generated from a policy *π*. A deterministic policy is a function *π* : S → A and a stochastic policy assigns a distribution *π*(.∣*s**t*) over the set of actions where actions are drawn from *a**t* ∼ *π*(*s**t*).\nE*π*[.] = E*a**t* = *π*(*s**t*)[.] respectively E*π*[.] = E*a**t* ∼ *π*(.∣*s**t*)[.] denote the expectation under the randomness of the state transition if the actions *a**t* of the agent are chosen according to policy *π* for deterministic or stochastic policies, respectively.\nThe *return* of a MDP starting at time index *t* is defined as\n \n*G**t* = ∑*k* = 0∞*γ**t* + *k**R**t* + *k* + 1, \nThe return satisfies the recursion, *G**t* = *R**t* + 1 + *γ**t* + 1*G**t* + 1.\nValue functions and action-value functions\n------------------------------------------\nFor a fixed policy *π*, the *value function* assigns to each state the expected return, conditional on starting from state and selecting actions according to policy *π* thereafter,\n \n*V**π*(*s*) = E*π*[*G**t*∣*s**t* = *s*]. \nMarkov property gives a fundamental recursive structure to the value function, the Bellman equation for the value function:\n \n$$\\begin{aligned}\nV^{\\pi}(s) = R\\_{t+1}+\\gamma\\sum\\_{s'}\\mathbb{P}^{\\pi}(s,s')V^{\\pi}(s'), \\label{bellman\\_eq}\\end{aligned}$$ \nwhere, P*π*(*s*, *s*ʹ) = P(*S**t* + 1 = *s*ʹ∣*s**t* = *s*, *a**t* ∼ *π*(*s**t*)).\nThe return of an MDP is a random variable, so one can take (conditional) expectations of its second moment:\n \n$$\\label{return\\_2nd\\_moment}\nM^{\\pi}(s) \\coloneqq \\mathbb{E}\\_{\\pi}\\left[G\\_t^2|s\\_t=s\\right].$$ \nIt is possible to formulate a Bellman equation for the expectation of the second moment of the return (derived in for a discounted infinite horizon with finite states and for the finite horizon in ) as follows: one has the expansion\n \n$$\\begin{aligned}\nG\\_t^2 &= (R\\_{t+1}+\\gamma G\\_{t+1})^2\\\\\n &= R^2\\_{t+1}+2\\gamma R\\_{t+1}G\\_{t+1}+\\gamma^2 G\\_{t+1}^2.\\end{aligned}$$ \nTherefore, one can re-write the expectation of the second moment of the return as,\n \n$$\\begin{aligned}\nM^{\\pi}(s) &= \\mathbb{E}\\_{\\pi}\\left[G\\_t^2|s\\_t=s\\right]= \n\\mathbb{E}\\_{\\pi}\\left[R^2\\_{t+1}+2\\gamma R\\_{t+1}G\\_{t+1}+\\gamma^2 G\\_{t+1}^2\\right]\\\\\n &= \\mathbb{E}\\_{\\pi}\\left[R^{(M)}\\_{t+1}\\right]+ \n \\mathbb{E}\\_{\\pi}\\left[\\gamma^2 M^{\\pi}(s\\_{t+1})\\right],\\end{aligned}$$ \nwhere,\n \n*R**t* + 1(*M*) = *R**t* + 12 + 2*γ**R**t* + 1*G**t* + 1. \nBy defining,\n \n$$r\\_{\\pi}^{(M)}(s) \\coloneqq\\mathbb{E}\\_{\\pi}\\left[R\\_{t+1}^{(M)}|s\\_t=s\\right]$$ \nthe Bellman equation for the expectation of the second moment takes the form,\n \n*M**π*(*s*) = *r**π*(*M*)(*s*) + ∑*s*ʹP*π*(*s*, *s*ʹ)*γ*2*M*(*s*ʹ). \nWe introduce another concept, closely related to the value function. The action-value function or the *Q-function* at state *s* and action *a* is the value of taking the action *a* and following the policy *π* after that,\n \n$$Q^{\\pi}(s,a) = \\mathbb{E}\\left[\\sum\\_{t=0}^T\\gamma^t R\\_{t+1}\\middle|s\\_0=s, a\\_0=a\\right].$$ \nThe value function can be expressed as the expectation of the Q-function across all possible actions,\n \n$$V^{\\pi}(s) = \\mathbb{E}\\left[Q^{\\pi}(s,a)\\middle|a\\_t\\sim\\pi(.|s)\\right].$$ \nThe Q-function satisfies the Bellman expectation equation,\n \n$$\\label{Q\\_Bellman\\_equation}\nQ^{\\pi}(s,a) = \\mathbb{E}\\left[ R\\_{t+1} + \\gamma Q^{\\pi}(s', a') \\;\\middle|\\; s\\_t = s, a\\_t = a\\right].$$ \nStandard reinforcement learning tries to find an optimal policy that maximizes the expected return, *π*\\* , *s**t**d* which would give the maximum action-value function *Q*(\\**s*, *a*) for any *s* and *a*,\n \n*Q*\\*(*s*, *a*) = *Q**π*\\* , *s**t**d*(*s*, *a*) ≥ *Q**π*(*s*, *a*). \nThere is a version of Bellman equation for the *Q*\\*-function:\n \n$$Q^{\\ast}(s,a) = \\mathbb{E}\\left[ R\\_{t+1} + \\gamma \\max\\_{a'} Q^{\\ast}(s', a') \\;\\middle|\\; s\\_t = s, a\\_t = a\\right]. \\label{bellman\\_optimality\\_equation}$$ \nSimilar to ([return2ndmoment]), one can define an action-value function for the expectation of the second moment as:\n \n$$K^{\\pi}(s,a) \\coloneqq \\mathbb{E}\\_{\\pi}\\left[G\\_t^2|s\\_t, a\\_t\\right]$$ \nand write that out as,\n \n$$\\label{second\\_moment\\_q}\nK^{\\pi}(s,a) = \\mathbb{E}\\left[ R\\_{t+1}^2 + 2\\gamma R\\_{t+1}Q^{\\pi}(s', a') + \\gamma^2 K^{\\pi}(s', a') \\;\\middle|\\; s\\_t = s, a\\_t = a\\right].$$ \nAs expected, the expectation of the second moment depends on that of the first moment. Therefore, to achieve a good approximation of the expectation of the second moment, a good estimate of the expectation of the first moment, i.e. the Q-function, is required. In section [ddpgsecond] we will use ([secondmomentq]) to find a policy that optimizes the second moment of Q-function. Similar ideas have been proposed by to optimize a combination of the return of the MDP and its variance.\nFinal quadratic hedging as Markov decision problem\n--------------------------------------------------\nOne way to fit quadratic hedging into an MDP/reinforcement-type learning set-up is to find such definitions of *γ* and *R**t* respective *R*(*s*, *a*, *s*ʹ) so that *G*0 = ∑*t* = 0*N**γ**t**R**t* + 1 = *L̃**T* or *G*0 = *L**T* (for deterministic interest rates, those two have the same optimizers). One typically uses *γ* = 1 or very close to 1 since *γ* cannot be independently chosen. One could try to learn strategies for smaller *γ* (or larger *γ*) and then move *γ* closer to 1. Some reinforcement algorithms only work (or can be proven to work) for *γ* < 1 and certain assumptions on rewards. We would then need a setting that satisfies those requirements. The methods we implemented here seem to work well enough for *γ* equal or very close to 1. Then, minimizing *G*02 will correspond to minimizing *L̃**T*2.\nWe thus set *γ* = 1, for *i* = 1, ⋯, *N* − 1 (with *T* = *t**N*)\n \n*R**i* = *H**t**i**S*(*S̃**t**i* + 1 − *S̃**t**i*) − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S̃**t**i*) \nand\n \n*R**N* = − TC(0, *H**T**S*, *S̃**T*) − *P**T*(*S̃**T*). \nIt is also possible to introduce some reference value function or process Ref with Ref*T* = *P**T*(*S̃**T*) and use it to “shift” rewards around:\n \n*R**i* = *H**t**i**S*(*S̃**t**i* + 1 − *S̃**t**i*) − TC(*H**t**i**S*, *H**t**i* − 1*S*, *S̃**t**i*) + (Ref*t**i* − Ref*t**i* − 1) \nwhere Ref*t*− 1 = 0. Ref*t* will telescope out and lead to the same *G*0 and *G*02 (and same optimizers) but the formulation with Ref might be more amenable to particular reinforcement learning approaches. We did not apply this idea here since we were able to run the implemented reinforcement learning approach without introducing such a function or process which also means that one does not need to compute or define it.\nIn, ([cashflowform]) is called the cash flow formulation, while ([accountingform]) is called the accounting formulation.\nMethods\n-------\nStandard reinforcement learning uses Bellman equations, together with temporal difference learning, and/or policy gradient lemma, in various types of methods. There are model based approaches in which the specification of the transitions and rewards are used and the sums or integrals are computed exactly or with some approximation so that all (or, at least, many) transitions and rewards are taken into account at once. However, this is only possible in some cases, often requires a finite set of states, and will often taken considerable computing power and memory. There are also various model free approaches that work based on observed transitions and rewards (single or a few observed steps each) rather than entire trajectories. Reinforcement learning algorithms try to learn value and action-value functions and a policy (or using a greedy or *ε*-greedy policy derived from the action-value functions). For standard reinforcement learning, these methods are discussed at many places, we only point to and work cited therein as well as for a treatment and discussion of reinforcement learning approaches and stochastic optimization approaches for a variety of settings. However, there are many introductions to reinforcement learning available. We will discuss reinforcement-type learning approaches that optimize the expectation of the second moment in the next section.\nThere are methods that work based on empirical/MC full-trajectory simulation of the sum of rewards, parametrizing the agent’s policy as deep neural networks, and optimizing the policy by taking gradient or stochastic gradient steps (or extensions thereof, such as ADAM). We call these methods deep trajectory based stochastic optimal control (DTSoC) methods. In these methods, one does not (need to) learn value or action-value functions. We will discuss such methods in the section after the next.\nIf the evolution of the environment as a system of SDEs and the controlled SDE can be captured in forward-backward stochastic differential/difference equations (FBSDE) and the objective function has appropriate initial, step-wise, and/or final terms, one can apply deep learning methods for FBSDE, which need limited model information and/or outputs but are otherwise generic. We will discuss such methods in a later subsection.\nReinforcement Learning\n======================\nWe refer to for a description of Q-learning, policy gradient methods, and deep deterministic policy gradient methods, to maximize the return of the MDP with standard reinforcement learning. One can implement a variant of Q-learning that learns both *Q* and *K* functions. Here, we will present, implement, and use a variant of deep policy gradients that learns a *Q* function, a *K* function, and a policy that minimizes the variance, i.e., minimizes the *K* function.\nDeep Deterministic Policy Gradients Variant for K-function\n----------------------------------------------------------\nIn this section, we will propose an algorithm which instead of optimizing *Q*-function optimizes *K**π*(*s*, *a*), the variance of the return rather than the expected return. Our proposed algorithm involves the same ideas as DDPG to approximate the *Q*-function. However, there are two major differences between our algorithm and the original DDPG algorithm: 1) The original algorithm includes one network to approximate the *Q*-function for the policy and one network to approximate the policy and it updates the parameters of the *Q*-network by minimizing the mean squared error of the difference between its approximation and target values. However, our algorithm includes an extra network *K**ω*(*a*, *s*) with set of parameters *ω* to estimate *K**π*(*s*, *a*). We update the parameters *ω* by minimizing the mean squared error between the output of the *K**ω* and its target values. The target values for the K-network are different from target values of *Q*-network and are generated using ([secondmomentq]). 2) The policy in the original DDPG is updated toward choosing the actions that optimizes the Q-function of the policy at a specific state *s*, however, in our algorithm we update the policy to choose the action that optimizes (minimizes) the K-function - the expectation of the second moment of the return. The details of our method are provided in Algorithm [mainalg]. See for a similar method that minimizes a combination of *Q* (mean) and *K* (variance) functions.\n[mainalg]\nInitialize policy parameters *θ*, Q-function parameters *ϕ*, K-function parameters *ω* Set target parameters equal to main parameters *θ*targ ← *θ*, *ω*targ ← *ω*, *ϕ*targ ← *ϕ* Initialize replay buffer M Initialize a random process N for exploration Observe initial state *s*1 Select action *a**t* = *μ**θ*(*s*) + N*t* Execute action, receive reward *r**t* and state transition *s**t* + 1 Append the transition to the replay buffer M Sample a mini-batch of transitions *B* = {(*s**i*, *a**i*, *r**i*, *s**i* + 1)} from the replay buffer M target*Q* = *r* + *γ**Q**ϕ*targ(*s*ʹ, *μ**θ*targ(*s*ʹ)) target*K* = *r*2 + *γ*2*K**ω*targ(*s*ʹ, *μ**θ*targ(*s*ʹ)) + 2*γ**r**Q**ϕ*targ(*s*ʹ, *μ**θ*targ(*s*ʹ)) **Q-update:** Get updated *Q*-function network weights *ϕ*updated by one-step gradient descent,\n \n$$\\nabla\\_{\\phi}\\frac{1}{|B|}\\sum\\_{(s, a, r,s')\\in B}\\left(Q\\_{\\phi}(s,a)-\\text{target}\\_{Q}\\right)^2$$ \n**K-update:** Get updated *K*-function network weights *ω*updated by one-step gradient descent,\n \n$$\\nabla\\_{\\omega}\\frac{1}{|B|}\\sum\\_{(s, a, r,s')\\in B}\\left(K\\_{\\omega}(s,a)-\\text{target}\\_{K}\\right)^2$$ \n**Policy update:** Get updated policy network weights *θ*updated by one"},"data_id":{"kind":"string","value":"arxiv_0182775"}}},{"rowIdx":45696,"cells":{"content":{"kind":"string","value":" the presence of the twist thus has an additional factor exp(*i**φ*/*N**s* ⋅ ∑*x**m**x*, 1) = exp(*i**φ* *w**s*[*m*]), where *w**s*[*m*] is the total spatial winding number of *m* in each configuration. In the definition of the spin stiffness *σ*, Eq. , the second derivative with respect to *φ* brings down − *w**s*[*m*]2 and setting *φ* to zero afterwards results in the expectation value one obtains\n \n$$\\begin{aligned}\n {\\sigma}=\\frac{N\\_s}{N\\_t}\\, \\langle w\\_s[m]^2\\rangle\n =LT\\, \\langle w\\_s[m]^2\\rangle\\;,\\end{aligned}$$ \nin the dual repesentation without twist (similar to *χ**n* in Eq. ).\nNumerical simulation, tests and basic analysis\n==============================================\nIn this section we collect several more technical aspects of this paper. We briefly discuss our simulation strategy for the dual formulation and evaluate the correctness of its results by comparing them to perturbative strong and weak coupling calculations. Furthermore we present numerical results for the phase diagram at finite volume and temperature as well as a finite volume scaling analysis which indicates that at nonzero temperature all transitions are smooth crossovers.\n![Energy density versus the coupling at \\mu=0. The analytical results for the strong and weak coupling expansions from Eq. agree very well with the numerical results (having very small error bars) obtained from simulating the dual ensemble on a 10\\times10 lattice and using Eq. .](Plots/o3_2dim_dual_worm_energy_strong_weak_vs_j_ \"fig:\") [figenergy]\nDual simulation and tests\n-------------------------\nIn this subsection we briefly discuss Monte Carlo simulation strategies for the dual representation Eq. ([eqdualZ]) and present tests for its correctness. Obviously each term in the partition sum ([eqdualZ]) is real and positive and a probability interpretation of the weights of the dual configurations is possible. The remaining challenge of a dual Monte Carlo simulation is to generate only those configurations that obey all the constraints. For the unconstrained dual variables $\\overline{m}\\_{x,\\nu}$ conventional local Metropolis updates are sufficient. For the constrained variables *m**x*, *ν* and *k**x*, *ν* the constraints enforce closed loops and loops where flux is conserved modulo 2, respectively. In both cases one can generate new admissible configurations by changing the variables along an arbitrarily chosen closed loop which guarantees that the constraints remain intact. This loop along which one updates the dual variables can for example be grown in steps using local random choices and corresponding Metropolis decisions until it closes, which is the well known worm strategy – this is the update used here. After equilibration, we typically use 105 to 106 measurements on configurations separated by ${\\mathcal O}(20)$ sweeps. The statistical errors we show are determined with the jackknife method taking autocorrelation times into account.\n\n![Top: Expectation value of the particle number density versus the chemical potential \\mu, both in lattice units, at different couplings J for lattice size 90 \\times 90. Bottom: Logarithm of the corresponding critical chemical potentials as a function of the coupling. We compare our data to the strong coupling result \\ln(3/J) and the weak coupling expansion of the mass gap and Eq. , having obtained C=102(2) from a fit of the J\\geq 1.4 data.](Plots/o3_2dim_dual_worm_mcrit_vs_j_ \"fig:\") [figpartvsmufirst]\nFor a first test we computed the energy density at vanishing chemical potential, cf. Eq. . In Fig. [figenergy] we show the results for *e* at *μ* = 0 as a function of the coupling *J* and compare to the weak and strong coupling expansions from Eq. . We find excellent agreement in the corresponding domains of *J*. This demonstrates that the mapping to the dual variables and the implementation of the dual Monte Carlo simulation are correct.\nWe now switch to the situation where the dual approach is really essential, i.e., the simulations at nonzero chemical potential. In the top panel of Fig. [figpartvsmufirst] we plot the particle density *n* measured via Eq. as a function of *μ*. As a general phenomenon at low temperatures a net density is induced into the system only after *μ* has reached a *threshold*, the mass of the lightest particle with charge coupling to *μ*. The critical values of *μ* visible in that plot thus depend on the coupling *J* just like the mass. This is shown in the bottom panel of Fig. [figpartvsmufirst], again with strong and weak coupling expansions. The former, *μ* = ln(3/*J*) for *J* → 0, is worked out in App. [appstrongcouplingmus], whereas the latter follows from the two-loop mass gap formula, Eq. , in which we obtain the constant *C* related to the UV cut-off by a one-parameter fit. The agreement is again very good, which is seen also in a comparison to the literature (Fig. 10 of ). Note also that the continuum scaling sets in at *J* ≃ 1.4, as for the energy density in Fig. [figenergy].\nMost of our lattice data were taken at *J* = 1.3, where *a**m* ∼ 0.22. This means a restriction to *μ* ≪ 5*m*, otherwise *a**μ* becomes comparable to 1 and strong discretization effects set in.\nFinite lattice phase diagram\n----------------------------\n![Phase transition in the J-\\mu plane for N_s=30 and N_t=1000. At large J, i.e., towards the continuum limit, the critical chemical potentials with increasing index (from red to green to black to blue) induce individual transitions between integer charges as utilized in. At small J the first N_s of them join to a single transition, \\mu_1=\\ldots=\\mu_{Ns}, increasing the particle density n in lattice units from 0 to 1 (see text). The next critical chemical potential \\mu_{N_s+1} (blue) is separated from those, and above it a density n=2 will be induced. At large J, on the other hand, the change from \\mu_{Ns} to \\mu_{N_s+1} plays no particular role. The strong coupling predictions for the transitions 0\\to 1 and 1\\to 2 are included as solid curves.](Plots/o3_2dim_dual_ns30_nt1000_mucritn_vs_j \"fig:\") [figmanymucs]\nBefore we come to discussing continuum results, we first consider nonzero lattice spacing and finite volume, i.e., we study the system at fixed lattice size without a final continuum limit *J* → ∞. We determine the *J*-*μ* phase diagram at low temperature, using the onset of non-zero particle density as a function of *μ* for determining the phase boundaries. Fig. [figmanymucs] shows this phase diagram for several critical chemical potentials. Note that the number of temporal sites is fixed, so the temperature in mass units *T*/*m* = 1/(*N**t**a**m*) varies with the coupling according to the mass gap formula *a**m*(*J*), e.g., Eq. .\nIn we have shown that close to the continuum limit at small temperatures and volumes, the particles are induced into the system one by one displaying integer plateaus in the particle number itself (not its density). These transitions show up for large *J* and some examples are plotted in Fig. [figmanymucs]. The locations of these critical *μ*’s are governed by the particle interaction phase shifts and thus contain interesting physics, as explained in detail in.\nIn the strong coupling regime at small *J* the situation is different. In the dual representation of the partition function, Eq. , every dual variable is suppressed by the corresponding power of *J*. Still, the temporal components of the flux variable *m* are promoted by factors of *e**μ*, that eventually overcome the factors of *J*/2. This mechanism acts locally on every spatial site, which means that if it is preferable to have *Q* particles at some site (*m**x*, 2 = *Q*), then this immediately applies to all sites. In fact, the superposition of *Q* fluxes does not cost any action at strong coupling. Therefore, in this regime the particle number *density* changes by one (in lattice units). The corresponding critical values of *μ* depend on *J* logarithmically, and their values are computed in Appendix [appstrongcouplingmus]. Fig. [figmanymucs] shows good agreement with these curves for small *J* and illustrates how intermediate couplings interpolate the transitions between these regimes. In particular all critical values of *μ* are on equal footing towards the continuum (large *J*), whereas in the strong coupling regime (small *J*) multiples of *N**s* bunch, such that regimes with fixed lattice density open up.\n![Phase transition as in Fig. [figmanymucs], but with a modified quantity on the y-axis (and some more data points for higher J’s), see text.](Plots/o3_2dim_dual_ns30_nt1000_mucritn_vs_j_final \"fig:\") [figmanymucso2]\nIn Fig. [figmanymucso2] we show the lattice phase diagram with the quantity *J*/2 ⋅ exp(*μ**Q*) on the *y*-axis. In the strong coupling limit *N**s* critical *μ*’s now meet at half integers 3/2, 5/2 etc., which agrees with the derivation in Appendix [appstrongcouplingmus] again. The corresponding diagram for the O(2) model is shown in Fig. 3 of Ref. (for this model the curves meet at integer values).\nLet us conclude this subsection with a quick look at the third dual variable *k**x*, *ν*, or more specifically at its sum *K* as defined in Eq. . For *μ* larger than the threshold the O(3) symmetry is explicitly broken to O(2) by the presence of the O(2) charge, which is expected to be manifest in a decrease of *K* (see the discussion below Eq. ). In Fig. [figthirddual] we show *K* as a function of *μ* (normalized by *μ**c*) and indeed find the onset of a drop at the critical chemical potential, which confirms the picture that the system tends to become more and more planar as *μ* increases.\n [figthirddual]\nCrossover at nonzero temperature\n--------------------------------\nWe conclude the first analysis of the lattice model by studying the nature of the transitions mapped out in the previous subsection. For this study we still keep the temperature *T* fixed and perform the standard scaling analysis of order parameters with the spatial volume *L* to analyze the nature of the transition.\n [figvolumescaling]\nFig. [figvolumescaling] shows the particle number density (top panel) and its susceptibility (bottom panel) at a fixed low temperature of 2% of the mass. As a function of chemical potential *μ* the particle number *n* is monotonically increasing, with a strong variation emerging only above *μ**c*, while the susceptibility displays a maximum. Doubling the spatial size, the data still fall on top of each other for both observables. Equivalent results were found for other nonzero temperatures and we conclude that for nonzero temperature the transition is smooth, i.e., it is a *crossover*.\nQuantum phase transition\n========================\nWe now explore the possibility of a quantum phase transition at zero temperature, i.e., we focus on the combined limit of zero temperature and infinite volume, i.e., *T* = 0, *L* = ∞ and analyze the transitions of the O(3) model as a function of the chemical potential. This amounts to sending the extents of both, Euclidean time and space to infinity. Of course, with finite numerical simulations this can only be done as a limit, *T* → 0 and *L*− 1 → 0. However, in general, the outcome depends on the particular choice of the trajectory towards that limit which one chooses in the *T*-*L*− 1 plane. In particular the spatial and temporal correlation lengths can be different, as expressed by the dynamical critical exponent (see Sec. [secqptstiff] below). Thus we will discuss the behavior of the system for different “scaling trajectories”, i.e., different paths leading to the limit *T* = 0, *L*− 1 = 0. For these scaling trajectories we will present and interpret our data for the particle density and the spin stiffness. For the interpretation we will partly rely on simple model systems showing characteristic features observed in our simulations.\nScaling trajectories towards zero temperature and infinite volume and particle number results\n---------------------------------------------------------------------------------------------\nAs already outlined, when considering the limits *T* → 0 and *L*− 1 → 0, the behavior of the system will depend on the particular scaling trajectory one follows in the *T*-*L*− 1 plane towards the origin. These limits can be taken in different ways, and a possible choice is to take the *L*− 1 → 0 limit first, keeping temperature fixed, and then take *T* → 0. We can also keep *L* fixed and take *T* → 0 first, and then *L*− 1 → 0. More generally we can take the limit *T*, *L*− 1 → 0 keeping [eq:scaling] TL= fixed, where *α* is a nonnegative real number. Generically, the constant has a noninteger mass dimension, for the practical implementation (e.g., for *α* = 2 below) one can fix (*T*/*m*)(*L**m*)*α* to a dimensionless constant. Notice that the consecutive limits mentioned above correspond to the limiting cases *α* = 0 and *α* = ∞. Here we will consider these two scaling trajectories plus *α* = 1, 2 as we illustrate in Fig. [figscalingdiagram]. In this subsection we briefly discuss these different scaling trajectories and partly describe the resulting physics using model calculations for the behavior at the emerging phase transitions.\n(0,0) circle (0.15); (0,0.3) – (0,4) – (3.2,4); (0,4.2) – (0,5); (0.3,0) – (4,0) – (4,3.2); (4.2,0) – (5,0); (0.3,0.3) – (3.4,3.4); (0.3,0.05).. controls (2.2,0.25) and (3,0.5).. (3.85,3.2); (T) at (-0.4,4.8) *T*; (L) at (5.5,0) *L*− 1; (a0) at (1.9,4.3) *α* = 0; (a1) at (1.4,2.4) *α* = 1; (a2) at (2.35,1.43) *α* = 2; (ainf) at (4.9,1.9) *α* = ∞;\n[figscalingdiagram]\nIn the limiting trajectories *α* = 0 and *α* = ∞ one has to perform the two limits consecutively. This is difficult to implement on the lattice but reveals interesting physics. Therefore, we will use simple model calculations to illustrate characteristic features of the consecutive limits. We will compare our numerical data to results for *free one-dimensional fermions*: At low densities the behavior is governed by the small momentum-exchange between the particles, which for the O(3) model at hand are one-dimensional repulsive bosons. At low momentum their behavior is universal and given by free (spinless) fermions, see Sec. [seccomparison].\nFor the other two scaling trajectories, *α* = 1 and *α* = 2 the results will not differ much concerning the particle density, but are characterized by a different behavior of the spin stiffness, which we will discuss in detail in Subsection [secqptstiff].\n \n \nThe *α* = 0 trajectory corresponds to the consecutive limits \n*L*− 1 → 0at fixed *T*, then *T* → 0 . \n The first limit of this sequence corresponds to the finite volume scaling at fixed nonzero temperature studied in Subsection [sectionfinitevolume]. This analysis has revealed a *crossover* as seen from Fig. [figvolumescaling]. Actually every fixed temperature possesses a specific crossover curve *n*(*μ*)*T*, which upon lowering the temperature *T* to zero in the second limit might become steeper near some *μ**c* and turn into a genuine phase transition. To some extent this feature is seen in our data shown in Fig. [figtempscaling] below, where it is, however, overlaid by the formation of “condensation steps” (see the discussion below).\n![Emergence of a phase transition in the limit T \\to 0 for free one-dimensional fermions in an infinite volume. We show that the smooth curves for T/m=1/10 (green) and T/m = 1/50 (blue) approach the T = 0 square root behavior from Eq. . This is a model calculation for the scaling trajectory \\alpha=0, especially its second limit T\\to 0 at L^{-1}=0.](Plots/dens_ferm_infL_Ttozero \"fig:\") [figfermroot]\nThe emergence of such a quantum phase transition as *T* approaches 0 is of course a well known feature which already appears in the simple model of free fermions in one spatial dimension: The density *n* is given by the Fermi-Dirac integral (*L*− 1 = 0),\n \n$$\\begin{aligned}\n n\\_f = \\int\\_{-\\infty}^\\infty\\!\\frac{dk}{2\\pi}\\,\\frac{1}{1+\\exp\\big((m+k^2/2m-\\mu)/T\\big)} \\;,\n \\label{eq\\_ferm\\_one}\\end{aligned}$$ \nwhere we use a nonrelativistic dispersion which is sufficient for a first illustration. The density can be expressed by a polylogarithm Li1/2, from which one obtains a square root in the *T* → 0 limit,\n \n$$\\begin{aligned}\n \\frac{n\\_f}{m} \\to \n &\\:\\frac{\\sqrt{2}}{\\pi}\\sqrt{\\mu/m-1}\n \\cdot\\Theta\\_{\\text{Heaviside}}(\\mu/m-1)\\notag\\\\\n &\\:\\mbox{for} \\quad L^{-1}=0,\\, T \\to 0 \\;.\n \\label{eq\\_ferm\\_two}\\end{aligned}$$ \nThe square root behavior, which is non-analytic at *μ*/*m* = 1, is indeed reached only for *T* = 0 via a sequence of analytic crossover-type curves as shown in Fig. [figfermroot].\n \n \nWe have already pointed out that our results for the density as a function of *μ* start to show “condensation steps” when the temperature *T* is lowered at fixed *L*. Thus it is interesting to study also the scaling trajectory *α* = ∞, which is defined by the consecutive limits\n \n*T* → 0at fixed *L*, then *L*− 1 → 0 . \nOur numerical results for the first one of these two limits are shown in Fig. [figtempscaling]. When lowering the temperature *T* at fixed *L* we observe the emergence of plateaus in the density where the particle numbers are integers. The plateaus correspond to sectors with fixed particle number, and are smoothed out by the finite temperature (the width of the transition region being proportional to the temperature). Therefore, sharp *steps* emerge in the *T* → 0 limit at any fixed *L*.\n [figtempscaling]\nAfter considering *T* = 0 at finite *L* for following the *α* = ∞ trajectory we need to perform the second limit *L*− 1 → 0. Also the behavior of this second limit can be illustrated in a one-dimensional fermion model, the Tonks-Girardeau limit of the Lieb-Liniger system, which we discuss in detail in Section [seccomparison]. One finds that the minimal energy in each particle number sector *Q* is *E*min(*Q*) = *π*2/(6*L*2*m*) ⋅ (*Q*2 − 1)*Q*. Steps occur whenever *E*min(*Q*) − *Q**μ* = *E*min(*Q* − 1) − (*Q* − 1)*μ* and thus are located at *μ**Q* = *m* + *π*2/2 ⋅ *Q*(*Q* − 1)/(*L*2*m*). The corresponding density *n*/*m* = 1/*L**m* ⋅ ∑*Q* = 1∞ΘHeaviside(*μ* − *μ**Q*) is shown in the top panel of Fig. [figstepslargervolumes]. It ‘oscillates’ around its square root limit reaching it by ever smaller oscillations. The square root can also be seen analytically, since for large *Q* the steps are at *μ**Q*/*m* = 1 + *π*2/2 ⋅ (*n*/*m*)2.\n\n [figstepslargervolumes]\nThe same behavior is seen in our numerical O(3) data shown in the bottom panel of Fig. [figstepslargervolumes], where, in the same fashion, ‘oscillations’ around a limiting curve diminish with increasing *L*. However, this can only be done at low but nonzero temperatures and, as discussed above, this gives rise to a crossover instead of a phase transition.\nIn the scaling trajectories *α* = 1 and *α* = 2 we consider approaches to the limit *T* = 0, *L*− 1 = 0 by sending *T* and *L*− 1 to zero simultaneously. The trajectories differ in their functional relation between *T* and *L*:\n \n$$\\begin{aligned}\n \\alpha=1: \\quad \n &T\\to 0 \\text{ and }L^{-1} \\to 0\\; \\text{ with }T = L^{-1}\\\\\n \\alpha=2: \\quad \n &T\\to 0 \\text{ and }L^{-1} \\to 0\\; \\text{ with }T \\propto (L^{-1})^2\\end{aligned}$$ \n [figdensitiesbothscalings]\nObviously *α* = 1 corresponds to square lattices *N**t* = *N**s*, while *α* = 2 corresponds to time elongated lattices with *N**t* ∝ (*N**s*)2. To be specific, in our numerical simulations for *α* = 2 we have used *N**t* = (*N**s*/2)2 at *J* = 1.3 (*a**m* ∼ 0.22), which amounts to *T*/*m* = 0.88/(*L**m*)2 or $Lm=0.94/\\sqrt{T/m}$.\nFig. [figdensitiesbothscalings] shows our simulation results for these two scaling trajectories. We show the particle density *n* as a function of *μ* for *T* → 0 and *L* → ∞ simultaneously. One can see in both trajectories that *n* vanishes for values of *μ* smaller than the threshold *m*, while it approaches certain nonzero values for *μ* > *m*. This is an indication for a nonanalyticity developing at *μ* = *m* in the limit *T* → 0, and thus for a *quantum phase transition*. Whether it emerges in the form of a jump in *n* (first order transition) or an infinite slope (second order) cannot be decided with our current data (and the common infinite volume limit *L* → ∞ to distinguish the orders is part of the scaling to zero temperature). The agreement with analytic predictions for repulsive bosons presented in Sec. [seccomparison] indicates a square root dependence on *μ* and thus a second order transition.\nMoreover, the data at lowest *T*/highest *L* do not differ much between *α* = 1 and *α* = 2. This bulk quantity therefore does not seem too sensitive to the particular scaling trajectory in the limit *T* → 0 and *L* → ∞.\nSpin stiffness and dynamical critical exponent\n----------------------------------------------\n [figstiffnessbothscalings]\nFig. [figstiffnessbothscalings] shows our spin stiffness data for the two scaling trajectories *α* = 1 and *α* = 2. Qualitatively, the stiffness behaves like the density: it vanishes for *μ* < *m* and is nonzero for *μ* > *m*. In contrast to the density, however, the stiffness shows a clear dependence on *α*: for *α* = 1 it is *significantly smaller* in the *μ* > *m* phase than for *α* = 2. In addition, the *stiffness agrees with the density* (divided by mass) for *α* = 2 at small densities, as Fig. [figstiffnessvsdensity] shows. Our stiffness data do not display the signature of a BKT transition for *μ* > *m* in the spatial correlations suggested in.\n [figstiffnessvsdensity]\nLet us now come to the interpretation of these data. It is known that the stiffness depends on the order of the limits *T* → 0 and *L* → ∞ : when *L* → ∞ is taken first (i.e. *α* = 0 scaling) the stiffness vanishes, whereas when *T* → 0 is taken first (i.e. *α* = ∞ scaling) it approaches the susceptibility of the ground state with respect to the same twist.\nFor *α* = 0 one keeps the temperature fixed (and thus the transition is a crossover), and then increases the spatial size to infinity. It is clear that in this way one looses all spatial correlations, the twist at the spatial boundary has no effect on the free energy for *L* → ∞ and the stiffness vanishes (the successive limit of zero temperature cannot change this any more). Our data for *α* = 1 suggest that this scaling trajectory is similar to *α* = 0 since the system size is growing too fast in comparison to *T* → 0, such that spatial correlations are lost leading to a *small stiffness*.\nFor *α* = ∞ one considers the zero temperature system at fixed volume and only afterwards the size is increased. At zero temperature one expects the ground state to determine the thermodynamic properties. In fact, the system at finite *L* is gapped proportional to 1/*L*2. Therefore, the partition function in the *T* → 0 limit is dominated by the ground state, and the free energy becomes the ground state energy (noting that the latter is *T*-independent, this is nothing but the third law of thermodynamics). As a consequence, the stiffness is the susceptibility of the ground state with respect to the spatial twist, as mentioned above.\nOn top of that, we now invoke an argument from the Bethe ansätze of Sec. [seccomparison]. One can easily convince oneself [3](#fn3) that the (nonrelativistic) pseudo-momenta *k**i* in the presence of a spatial twist *φ* are shifted by *φ*/*L*. The ground state energy ∑*i* = 1*Q**k**i*2/(2*m*) receives no linear term in *φ* since the total momentum ∑*k**i* vanishes. The quadratic term is the constant *Q* ⋅ (*φ*/*L*)2/(2*m*), and we immediately obtain the stiffness as *σ* = *Q*/(*L**m*) = *n*/*m*. This explains that the *stiffness equals the density* for *α* = ∞, for small densities. Our data for *α* = 2 suggest that this scaling trajectory is similar to *α* = ∞ since the stiffness still equals the density, as seen in Fig. [figstiffnessvsdensity].\nFinally, let us discuss the *dynamical critical exponent* *z* that results from our findings. The correlation length near a second order phase transition is known to diverge. To achieve *T* = 0 in the quantum phase transition one has to send the extent of the Euclidean time *β* to infinity (just what we performed in the lattice simulations), and only at *β* = ∞ the phase transition occurs and the spatial correlation length *ξ* is infinite. However, the system also has a typical correlation length *ξ**τ* in the Euclidean time direction. If nothing in the system distinguishes between space and Euclidean time, the two correlation lengths have to agree (which corresponds to *z* = 1 below). However, in general – and in particular in the presence of *μ* which breaks the Euclidean symmetry by coupling to temporal components/winding numbers – the two correlation lengths are related by *ξ**τ* ∼ *ξ**z* with *z* the dynamical critical exponent. Note the similarity to our scaling trajectories *β* ∼ *L**α* and indeed our stiffness data can be used to determine *z* and the correlation length critical exponent *ν*. For that we use finite size scaling of the free energy density,\n \n$$f\\sim\\frac{1}{L\\beta}\\,g\\big(\\frac{\\xi}{L},\\frac{\\xi^z}{\\beta};\\varphi\\big)$$ \nwhere the prefactor is the inverse volume (in 1 + 1 dimensions) making *f* = *F*/*L* an intensive quantity and *g* is a universal function of the spatial correlation length, the box sizes and the twist angle. Again, noninteger dimensions can be compensated by the corresponding powers of the mass. Using the scaling *ξ* ∼ *δ*− *ν* with the normalised distance to the critical value *δ* = *μ*/*m* − 1, and the definition we get for the stiffness\n \n*σ* ∼ *L*1 − *z* *h*(*L*1/*ν**δ*, *T**L**z*) \nwhere *h* is another universal function.\n [fig:finitesizefirst]\nFor practical purposes one can derive two criteria :\n \n$$\\begin{aligned}\n (a) &\\text{ the curves }\\mu\\text{ vs.\\ }L^{z-1}{\\sigma}\\text{ intersect at }\\mu=m,\\\\\n &\\text{ if }TL^z\\text{ is kept constant}\\end{aligned}$$ \nsimply because at *δ* = 0 the length *L* does not enter as argument of *h* anymore and\n \n$$\\begin{aligned}\n (b) &\\text{ the curves }L^{1/\\nu}\\delta\\text{ vs.\\ }L^{z-1}{\\sigma}\\text{ collapse to a single curve,}\\\\ \n &\\text{ again if }TL^z\\text{ is kept constant}\\end{aligned}$$ \nOur data are collected at constant *T**L**α* with *α* = 1, 2 and thus we can ask whether *z* = 1 or *z* = 2 obey these statements, at least approximately. We will rescale all dimensionful quantites by the corresponding power of mass.\nFor statement (a) with *z* = 1 we thus plot *σ*(*μ*) keeping *T**L* constant, i.e., on the scaling trajectory *α* = 1, while for *z* = 2 we plot *L*1*σ*(*μ*) keeping *T**L*2 constant, i.e., on the scaling trajectory *α* = 2, and see whether the data for different *L*’s intersect. As Fig. [fig:finitesizefirst] shows, the curves from our numerical data intersect approximately for *z* = 2 (albeit at *μ* slightly smaller than *m*), whereas for *z* = 1 they certainly do not intersect.\n [fig:finitesizesecond]\n [fig:finitesizethird]\nFor statement (b) one has to deal with the appearance of another critical exponent *ν*. In Fig. [fig:finitesizesecond] we plot the obervables of this statement assuming the value *ν* = 1/2 for free fermions. Again a collapse to a single describes the situation much better for *z* = 2. Finally, we can assume *z* = 2 and check the value of *ν* away from 1/2. Fig. [fig:finitesizethird] shows data proposing *ν* = 1/3 and *ν* = 3/4, for which the curves certainly do not collapse.\nFrom this analysis we conclude that the critical exponents of the O(3) quantum phase transition at *μ* = *m* are consistent with *z* = 2 and *ν* = 1/2, the values of free 1d fermions.\nComparison to analytic results on repulsive bosons\n==================================================\nIn this section we will compare our density data to the density for systems of repulsive bosons in one spatial dimension. One of the best known examples is the one introduced by Lieb and Liniger (LL), where boson pairs interact via a Dirac delta function. The system can be solved in terms of plane waves picking up phase shifts *δ* when two bosons are interchanged. Such a Bethe ansatz works for nonrelativistic or relativistic bosons, without antibosons though.\nThe phase shifts of the O(3) model are also known and those of “isospin 2”,\n \n$$\\begin{aligned}\n \\delta=-\\arctan\\frac{\\pi}{2\\theta}~,\\end{aligned}$$ \nwith the relative rapidity,\n \n$$\\begin{aligned}\n \\theta=\\text{arsinh}(k/m) \\;,\\end{aligned}$$ \ngovern the low density regime. We will thus use the Bethe ansatz equations with those phase shifts. Note that the O(3) wave functions are not plane waves anymore, but the Bethe ansatz is believed to give an exact result at low densities.\nAt very low momenta the O(3) phase shifts agree with those of the LL model, actually all repulsive one-dimensional bosons are universal in the deep IR, where the precise UV shape of the interaction is not relevant and where *δ*(0) = − *π*/2 by Levinson’s theorem. Using only this value one arrives at the Tonks-Girardeau (TG) gas, which is the infinite coupling limit of the LL model. Its eigenvalues are known to be that of free fermions with the modification that the numbers *I**i* in Eq. below are not always half-integers as is the case for antiperiodic fermions (which is not relevant for large *L*) and that the eigenfunctions of the TG gas are still symmetric under exchange of bosons. We have already shown that this results in the square root behavior of the particle density for *μ* ≈ *m* at zero temperature and infinite volume (see also Eq. below).\nTo be more precise, the relativistic Bethe ansatz reads\n \n$$\\begin{aligned}\n Lm \\sinh(\\theta\\_i) -2\\sum\\_{\\substack{j=1\\\\j\\neq i}}^Q\\Delta(\\theta\\_j-\\theta\\_i)=2\\pi I\\_i~,\n \\label{eq\\_bethe\\_ansatz}\\end{aligned}$$ \nwhere the *I**i* are distinct, and half-integer/integer for even/odd total charge *Q*. In our conventions the dynamics enters via the phase shifts in [eq:Delta] =+/2 with lim*k* → 0Δ = 0. This ansatz works for ground states (with the choice {*I**i*} = { − (*Q* − 1)/2, …, (*Q* − 1)/2}) and excited states at fixed particle number *Q* in any finite volume *L*. The nonrelativistic Bethe ansatz can be obtained by approximating the rapiditites as *θ* ≈ *k*/*m*. Both are not too hard to solve numerically.\n \n [fig:latticecompare]\nFurthermore, Yang and Yang have derived a thermodynamic Bethe ansatz for the density *ρ*(*k*) and energy density *ε*(*k*) at nonzero temperatures and chemical potentials in infinite volumes. The non-relativistic Yang-Yang equations give rise to\n[eq:YYnonrel]\n \n$$\\begin{aligned}\n 2\\pi\\rho(k)[1+e^{\\epsilon(k)/T}]\n &=1+\\int\\_{-\\infty}^\\infty\\!d\\tilde{k}\\,\n \\rho(\\tilde{k})\\,\\Delta'(k-\\tilde{k})~,\\\\\n \\epsilon(k)\n &=\\frac{k^2}{2m}+(m-\\mu)-\\frac{T}{\\pi}\\\\\n &\\times\\!\\!\\int\\_{-\\infty}^\\infty\\!\\!\\!d\\tilde{k}\\,\n \\log\\big(1+e^{-\\epsilon(k)/T}\\big)\\,\n \\Delta'(k-\\tilde{k})\\,.\\notag\\end{aligned}$$ \nwhile, following similar arguments of, and ignoring thermal pair production[4](#fn4), the relativistic Yang-Yang equations read\n[eq:YYrel]\n \n$$\\begin{aligned}\n 2\\pi\\rho(\\theta)\\big(1+e^{\\epsilon(\\theta)/T}\\big)\n &=m\\cosh\\theta+2\\int\\_{-\\Theta}^{\\Theta}d\\tilde\\theta\\;\\rho(\\theta)\\Delta'(\\theta-\\tilde\\theta)\\;,\\\\\n \\epsilon(\\theta)\n &=m\\cosh\\theta-\\mu\n -\\frac{T}{\\pi}\\\\\n &\\times\\!\\int\\_{-\\Theta}^{\\Theta}\\!\\!d\\tilde\\theta\\;\\log(1+e^{-\\epsilon/T})\\Delta'(\\tilde\\theta) \\;.\\notag\n \\end{aligned}$$ \nIn both relativistic and non-relativistic equations, $\\Delta'(x)=\\frac{d\\Delta(x)}{dx}$, where Δ is defined by. Having solved this system one can extract the particle density as *n* = ∫− ∞∞ *d**k̃* *ρ*(*k̃*).\nFinally, the zero temperature limit for the density is,\n \n$$\\begin{aligned}\n 2\\pi\\rho(k)\n &=1+\\int\\_{-K}^K\\!d\\tilde{k}\\,\n \\rho(\\tilde{k})\\,\\Delta'(k-\\tilde{k})~,\\quad \\rho(|k|>K)=0~,\\end{aligned}$$ \nwhere *K* is the analogue of the Fermi momentum related to *μ* by $K=\\sqrt{2}\\,m\\sqrt{\\mu/m-1}$. Low densities amount to *μ* ≈ *m* and thus *K* → 0, for which the integration range shrinks, *ρ*(0) = 1/(2*π*) and *n* = *ρ*(0) 2*K* such that\n \n$$\\begin{aligned}\n \\frac{n}{m}\n =\\frac{\\sqrt{2}}{\\pi}\\, \\sqrt{\\mu/m-1}\n \\qquad(T=0,\\text{ small }n/m)~,\n \\label{eq\\_square\\_root\\_again}\\end{aligned}$$ \nconfirming once more the square root behavior for low densities.\nWe solve these integral equations numerically and compare to the lattice data in Fig. [fig:latticecompare]. The agreement is fairly good. Note that these are parameter-free ansätze, so no fit is involved in the comparison. Let us repeat that the agreement is expected for low temperatures only, since Bethe ansätze do not contain antibosons and pair production.\nSummary\n=======\nIn this paper we have presented thermodynamic lattice simulations of the asymptotically free two-dimensional O(3) model. The complex action problem of the conventional representation at finite chemical potential is overcome by using a representation in terms of dual variables, i.e., we simulated worldlines. In the O(3) model the mass of the particle triplet is generated dynamically, and in our simulations we have confirmed the expectation that at low temperatures a nonzero particle number (or charge) density occurs only when *μ* reaches the threshold given by the mass. Furthermore, in small volumes several critical values of *μ* occur which correspond to integer particle numbers induced by *μ* (for finite lattice couplings see the lattice phase diagram in Sec. [seclatticediagram]).\nA finite volume scaling analysis has revealed that at nonzero temperatures the particle density as a function of *μ* is regular, i.e., the transition is a crossover. The possibility of a phase transition at zero temperature has been analyzed through a simultaneous scaling of Euclidean time and space. Indeed, the data indicate a quantum phase transition at *μ* = *m*, which, using the agreement with Bethe ansätze for one-dimensional repulsive bosons, should be of second order because the particle density follows the universal square root behavior. Further comparisons to these ansätze, including a nonzero temperature, have shown that the O(3) model can indeed be described by these bosons. A more detailed analysis of this finding should include the continuum limit of our lattice data (which we believe will not change the results significantly, as we have observed for some observables at lattice coupling *J* = 1.4), but is beyond the scope of the present study.\nWe have also measured the spin stiffness which in the conventional representation is defined via twisted spatial boundary conditions and in the dual representation measures spatial winding numbers of the worldlines. Based on the boson description at zero temperatures with only ground states, the stiffness is expected to be equal to the particle density. This is consistent with our low temperature data, when scaling *L* such that *T**L*2 is constant, i.e., *α* = 2. However, when *L* is larger in the zero temperature limit, e.g., when *T**L* is kept constant, i.e., *α* = 1, then this equality does not hold and the spin stiffness is significantly lower, which indicates a lost correlation between the spatial boundaries. From these findings we conclude that the dynamical critical exponent *z* is close to 2, in agreement with free fermions.\nAlthough originally introduced for studying a potential BKT transition, our stiffness data have not indicated such a transition in the O(3) system at chemical potentials larger than the mass where it tends to be planar (as we have also confirmed).\nBesides higher precision, further studies in various directions would be useful: First of all, perturbation theory should match our lattice data at high *μ*’s (keeping *a**μ* small to avoid discretization effects). Secondly, measuring the vortex correlation functions or the topological susceptibility should shed light on the possibility of a BKT-like transition. Extensions to higher O(N) or CP(N-1) models as well as to 2+1 dimensions could be done straightforwardly with the dual representation.\nThe physically very interesting regimes of complex chemical potentials, where fractional instantons should occur (which in turn underly the resurgence program mentioned in the introduction), and at large theta angle, with the Haldane conjecture of a phase transition at theta angle equal to *π*, remain a challenge as the dual variables used here do not solve the sign problem in these situations.\nWe thank Quirin Hummel, Yannick Meurice, Lode Pollet, Klaus Richter, Juan Diego Urbina, Andreas Wipf and in particular Hans-Gerd Evertz for helpful discussions. FB is supported by the DFG (BR 2872/6-1) and TK by the Austrian Science Fund, FWF, DK *Hadrons in Vacuum, Nuclei, and Stars* (FWF DK W1203-N16). Furthermore this work is partly supported by the Austrian Science Fund FWF Grant. Nr. I 1452-N27 and by the DFG TR55, *”Hadron Properties from Lattice QCD”*. TS acknowledges the support of the DOE grant E-SC0013036.\nCritical *μ*’s at strong coupling\n=================================\nIn this appendix we derive the critical values of the chemical potential in the strong coupling limit which we use for comparing to our numerical data in Figs. 2 and 3.\nFor small values of *J*, the configurations that dominate the partition function *Z* of Eq. have minimal values of all dual variables,\n \n$$\\begin{aligned}\n \\quad k\\_{x,\\nu}=\\overline{m}\\_{x,\\nu}=0 \\quad \\forall x, \\nu \\quad \\mbox{and} \\quad m\\_{x,1} = 0\n \\quad \\forall x~,\\end{aligned}$$ \nexcept the temporal component of the flux variable, which assumes a constant value\n \n$$\\begin{aligned}\n m\\_{x,2}={r}\\quad \\forall x~,\\end{aligned}$$ \nwhich amounts to *r* static particle world lines on each temporal bond. From Eq. it is clear that the resulting particle density is *n* = *r* (in lattice units). We restrict ourselves to positive *μ* and thus positive *r* for simplicity. These configurations obey the constraints in Eq. and result in the partition functions\n \n$$\\begin{aligned}\n Z\\_{r}(J,\\mu)=\\left[\\left(\\frac{Je^{\\mu}}{2}\\right)^{r}\\,\\frac{\\Gamma(1/2)}{\\Gamma(r+3/2)}\\right]^{N\\_sN\\_t},\\end{aligned}$$ \n(where the factorials have cancelled against Γ-factors from the beta function) or grand potential densities\n \n$$\\begin{aligned}\n \\Omega\\_{r}(J,\\mu)=-{r}\\ln\\big(\\frac{Je^{\\mu}}{2}\\big)+\\ln\\Gamma({r}+3/2) \\;,\\end{aligned}$$ \nup to an irrelevant additive constant. Certain values of *r* yield the smallest grand potential Ω*r* depending on the values of *μ* and *J*. 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Zamolodchikov, Relativistic Factorized S Matrix in Two-Dimensions Having O(N) Isotopic Symmetry, Nucl. Phys. **B133** (1978) 525; JETP Lett. **26** (1977) 457.\nB. Sutherland, *Beautiful Models, 70 Years of Exactly Solved Quantum Many-Body Problems*, World Scientific (2004).\nN. Levinson, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. **25** (1949) 9.\n---\n1. The O(2) lattice action is − *J*∑*x*, *ν*cos(*ϕ*(*x* + *ν̂*) − *ϕ*(*x*)), which is nothing but with *r*3 = 0, *r*12 = 1 at *μ* = 0.[↩](#fnref1)\n2. This is obtained from *e* = ⟨*E*⟩/*N**s**N**t*, where *E* = ∑*x*, *ν*(∇*ν**r⃗*)2 and (∇*ν**f*)(*x*) = *f*(*x* + *ν̂*) − *f*(*x*). From the normalization of *r⃗* it follows that *e* = 4 − 2⟨*r⃗*(*x*)*r⃗*(*x* + *ν̂*)⟩/*N**s**N**t* and thus.[↩](#fnref2)\n3. The *I**i* in Eq. come from closing the boundary and thus need to be shifted by *φ*/(2*π**L*). Moreover, it is sufficient to consider the nonrelativistic Bethe ansatz and thus replace *θ* ≈ *k*/*m* ≈ sinh(*θ*). Then the *I**i*-shift can be transfered to a uniform shift in the *k**i*, because the Δ-term contains differences of *k*’s only and thus is unchanged.[↩](#fnref3)\n4. Because of this, the equations are only exact at zero temperature. Nevertheless we will use them at finite, but small temperature to compare to lattice data.[↩](#fnref4)\n\nThe two-dimensional O(3) model at nonzero density:from dual lattice simulations to repulsive bosons\n===================================================================================================\nWe discuss the thermodynamics of the O(3) nonlinear sigma model in 1+1 dimensions at nonzero chemical potential (equivalent to a magnetic field). In its conventional field theory representation the model suffers from a sign problem. By dualizing the lattice model we present, for the first time, nonzero density data of an asymptotically free theory with dynamical mass-gap. We find a quantum phase transition at zero temperature where as a function of the chemical potential the density assumes a nonzero value. Measuring the spin stiffness we present evidence for a corresponding dynamical critical exponent *z* close to 2. The low energy O(3) model is conjectured to be described by a massive boson triplet with repulsive interactions. We confirm the universal square root behavior expected for such a system at low density (and temperature) and compare our data to the results of Bethe ansatz solutions of the relativistic and non-relativistic one-dimensional Bose gas. We also comment on a potential Berezinskii- Kosterlitz-Thouless transition at nonzero density.\nIntroduction\n============\nAnalyzing quantum field theories at nonzero temperature and chemical potential is of interest not only for studying their thermodynamical properties, but may also provide deep insights into the physical structure of the theory under consideration. A recent such example is given in, where we demonstrated that the grandcanonical ensemble at low temperature and small volumes can be connected to scattering data.\nBefore we outline the physics results presented in this paper, we briefly mention the challenges of thermodynamics for numerical lattice simulations, one of our best ab-initio tools (since analytic approaches are typically limited). Monte Carlo simulations are based on importance sampling and, while finite temperature simulations are routine, simulations at nonzero chemical potentials *μ* are in many cases plagued by the complex action/sign problem: at nonzero *μ* the action *S* has a non-vanishing imaginary part and the Boltzmann factor *e*− *S* cannot be used as a probability weight in a stochastic process. One of the most successful methods is to employ dualization of the lattice path integral to new, so-called ‘dual variables’, where the partition sum has only real and positive contributions (see, e.g., for reviews). Although, it is not completely clear for which classes of models a dual representation is useful for treating the sign problem, in models where it is successful the dual representation has allowed the exploration of the finite density phase diagrams, and the corresponding data also serve as a benchmark for other approaches to the complex action problem.\nWe would like to stress that dual representations not only enable simulations at nonzero density, but also reveal physical aspects complementary to those of the standard representation. In the dual representation the dynamical degrees of freedom are worldlines and the chemical potential couples to their temporal component. Via a discrete version of current conservation on a space-time lattice one finds that the chemical potential indeed couples to the temporal winding number of the worldlines. Thus the net number of particles (charged under the Noether charge of the corresponding symmetry) can be identified with the temporal net winding number of the worldlines and the Noether charge becomes topologically conserved. As such the worldlines winding around the time direction have a direct interpretation as quantum states carrying the corresponding quantum number. Since this is an all-scale statement, the dual worldlines carry direct information about the infrared physics, generally obscured in the conventional formulation of asymptotically free theories.\nThe O(3) nonlinear sigma model, which we are dealing with in this work, is conjectured to possess a particle triplet as its infrared excitations, the mass of which is generated dynamically. The dual wave function method of indeed utilizes the spatial distance of the worldlines to obtain information about the particles’ scattering. In the case of Quantum Chromodynamics (QCD), which is also asymptotically free, the low energy excitations are not the colored quarks and gluons, but colorless hadrons, with a large fraction of their masses being dynamically generated. A related dual lattice representation is that in terms of meson hoppings plus (anti-)baryon worldlines, to which the baryon chemical potential couples (this dual representation, however, does not remove the sign problem completely and does not take into account the gauge action).\nTo be more concrete, we introduce a chemical potential for an O(2) subgroup of the O(3) symmetry of the sigma model in two dimensions. At low temperatures a nonzero density is expected to be induced into the system when *μ* reaches the threshold of the particle mass. In the condensed matter context *μ* can be viewed as a constant magnetic field inducing a magnetization; see Secs. 11 and 19 of. We will analyze this transition in detail extending our previous simulations to large volumes. Beyond the transition the system explicitly breaks the global internal symmetry from *O*(3) to *O*(2) and in principle allows classical vortex solutions with half-integer topological charge. Although at asymptotically large densities vortices are expected to bind in neutral pairs, it was conjectured in that these vortices will be liberated at small densities, so that a Berezinskii-Kosterlitz-Thouless (BKT) transition may happen at some finite value of the density.\nOur main findings are twofold: Concerning the O(3) phase diagram we find a threshold *crossover* at nonzero temperatures and a *quantum phase transition* at zero temperature, as a function of *μ* at the mass threshold. Note that lattice simulations become expensive in this limit, as both temporal and spatial extension must be taken to infinity. Using the concept of spin stiffness we analyze spatial correlations and present evidence for a dynamical critical exponent *z* close to 2, which is consistent with the non-relativistic free fermions to which a model at low density belongs to. We do not directly see signatures of a BKT transition conjectured in in any of the observables studied in this work.\nSecondly, the numerical data for the density near the transition can be described by a square root. The latter is universal for *one-dimensional repulsive bosons*. The simplest example of which is the non-relativistic quantum particle gas with repulsive delta-function interactions (the Lieb-Liniger model ) and its limit when the repulsion strength goes to infinity (the Tonks-Girardeau limit ). The latter is equivalent to free fermions. These systems only differ in the specific form of the phase shifts, relevant away from the transition. Eigenstates and thermodynamics of these systems can be obtained from Bethe ansätze. We show that our data match well with the corresponding analytical nonrelativistic or relativistic predictions. Our simulations are performed at a lattice coupling in the continuum scaling regime, and we believe that the continuum limit to be performed does not reveal new qualitative features.\nOur study demonstrates that dual lattice simulations are capable of describing a system all the way from the Lagrangian in terms of ultraviolet fields to the infrared physics in terms of interacting particles (at nonzero densities induced by *μ*). Such a transformation is the essence of Wilson’s renormalization group, which is probably very hard to tackle analytically, but specific questions can be answered by the lattice, now that we have reliable lattice simulations at hand. To better understand the structure of dual partition functions and observables should also be of help in this program.\nWe briefly mention at the end that, although this program is mostly inspired by the attempts to understand the nonzero density structure of QCD, the potential benefit of studying nonlinear sigma models goes past this problem. Firstly, they are interesting in itself and appear as effective models of (anti-) ferromagnetic systems. Secondly, the study of properties of these systems is interesting in the context of continuum quantum field theories themselves. On the one hand, we have recently shown for the example of the O(3) nonlinear sigma model that lattice dualities may provide a physical connection with the low energy excitations of the theory. On the other hand, the nonlinear sigma models play a crucial role in the development of the continuum definition of quantum field theories via the resurgence program (see and references therein). To date, resurgent constructions were explicitly shown to work only in analytical tractable one-dimensional reductions of the O(N) and CP(N) nonlinear sigma models as well as of the principle chiral model. However, genuinely 1+1 dimensional nonzero density systems, akin to what we study here, also show similar resurgence structures. Numerical and physical understanding of these systems is therefore important for the fundamentals of quantum field theory as well.\nDefinition of the model and its lattice discretization\n======================================================\nContinuum formulation and observables\n-------------------------------------\nThe O(3) model is conventionally written in terms of normalized vectors *r⃗*(*x*) = (*r*1(*x*), *r*2(*x*), *r*3(*x*)) with *r⃗*(*x*)2 = 1 ∀*x* (also called ‘spins’ or ‘O(3) rotors’) and the continuum action reads,\n \n$$\\begin{aligned}\n S[\\vec{r}\\,] \\; \n = \\; \\frac{1}{g^2} \\int \\! d^2x \n &\\Big[ \\, \\frac{1}{2}\\, \\big (\\partial\\_\\nu r\\_a \\big)^2 + \n i\\,\\mu\\,\\big(r\\_1\\partial\\_2 r\\_2-r\\_2 \\partial\\_2 r\\_1\\big)\n \\notag\\\\\n &+\\frac{\\mu^2}{2}\\,r\\_3^2-\\frac{\\mu^2}{2} \\, \\Big]~,\n\\label{eq:def\\_action\\_cont} \\end{aligned}$$ \nwhere the coupling constant *g* is dimensionless in 1+1 dimensions. We have already coupled a chemical potential *μ* to one of the O(2) subgroups, which excites the 3-component of the angular momentum. Repeated indices are summed over (*ν* = 1, 2 and *a* = 1, 2, 3) and arguments *x* have been dropped. At nonzero temperature *T* the Euclidean time *x*2 is periodic with period 1/*T*. In such a bosonic theory, *μ* also enters quadratically tending to suppress the perpendicular component *r*3(*x*).\nOur main thermodynamic observables will be the expectation values of the charge *Q*, its density *n* and its susceptibility *χ**n*:\n \n$$Q\n =T\\,\\frac{\\partial \\ln Z}{\\partial \\mu}~,\\qquad\n n\n =\\frac{Q}{L}~,\\qquad\n \\chi\\_n \n = \\frac{\\partial n}{\\partial \\mu}~,$$ \nwhere *Z* is the grand canonical partition function (see or Eq. below). Eventually, all dimensionful quantities like *μ*, *T*, *L*, *n* etc. will be given in units of the mass *m*, e.g.,\n \n$$\\frac{n}{m}\n =\\frac{T/m}{Lm}\\,\n \\frac{\\partial \\ln Z}{\\partial (\\mu/m)}~,$$ \nwhereas *χ**n* is already dimensionless.\nWe will also explore the spin stiffness for which one imposes twisted spatial boundary conditions for a finite spatial length *L*. To implement these, we first introduce the O(2) polar angle *ϕ* by combining the first two components into a complex number\n \n*r*1(*x*) + *i**r*2(*x*) = *r*12(*x*)*e**i**ϕ*(*x*) , \nand then replace the periodic boundary conditions in space by twisted ones,\n \n*ϕ*(*x*1 + *L*, *x*2) = *ϕ*(*x*1, *x*2) + *φ* . \nIf the twist costs free energy, *F* = − *T*ln*Z*, at leading order the dependence of *F* on *φ* is quadratic in *φ*, and we define the *spin stiffness* (also called superfluid density) by\n \n$${\\sigma}=L\\,\\left.\\frac{\\partial^2 F}{\\partial{\\varphi}^2}\\right|\\_{{\\varphi}=0}\n =-L\\,T\\,\\frac{1}{Z}\\,\\left.\\frac{\\partial^2 Z}{\\partial{\\varphi}^2}\\right|\\_{{\\varphi}=0}~,\n \\label{eq\\_def\\_stiff}$$ \nwhere we have used that *Z* is an even function of *φ*. Physically it is clear that *σ* depends on whether the regions *x*1 and *x*1 + *L* are correlated, i.e., whether the system is in a (spatially) ordered state.\nThe spin stiffness can be computed and related to vortices in the lattice O(2) model without chemical potential[1](#fn1). For large lattice coupling, the vortices arrange in pairs and the spin correlator decays algebraically, which is the behavior closest to an ordered state in two dimensions (as the Mermin-Wagner theorem forbids the spontaneous breaking of the continuous symmetry). As a consequence, the spin stiffness *σ* will be nonzero. At small lattice coupling, the vortices condense and make the correlator decay exponentially. In this regime the spin stiffness *σ* will vanish if *L* is larger than the spatial correlation length *ξ*. This is why the spin stiffness can be used to detect BKT transitions characterised by the change of the correlator decay and to measure the spatial correlation length.\nActually, the dimensionful combination *φ*/*L* may be viewed as an imaginary chemical potential in the spatial direction, and therefore the stiffness is known to measure spatial winding numbers. As this is best seen using dual variables, we give the corresponding formula in the next section.\nLattice formulation and the dual representation\n-----------------------------------------------\nThe lattice action discretizing reads,\n \n$$\\begin{aligned}\n S[\\vec{r}\\,] \n =& - J\\sum\\_{\\substack{x\\in\\Lambda\\\\ \\nu=1,2}}\n \\Big[ r\\_{3}(x)\\, r\\_{3}(x+\\hat{\\nu})+ \\frac{1}{2} \\, r\\_{12}(x) \\, r\\_{12}(x+\\hat{\\nu})\n \\notag\\\\\n &\\times\\big\\{e^{-i(\\phi(x)-\\phi(x+\\hat{\\nu}))-\\mu\\,\\delta\\_{\\nu,2}} + \n c.c.|\\_{\\mu\\to -\\mu}\\big\\}\\Big] \\;.\n \\label{eq\\_lattice\\_action}\\end{aligned}$$ \nAs common in lattice field theory, the chemical potential *μ* introduces exponential factors for the forward and backward temporal hopping terms. For *μ* = 0 these terms are related by complex conjugation *c*.*c*., but when *μ* has a nonzero real part we face a complex action problem. The parameter *J* is the lattice coupling constant (dimensionless and positive) and the first sum runs over the *V* ≡ *N**s* × *N**t* sites of a two-dimensional lattice Λ with periodic boundary conditions. Again *ν* = 1, 2 and *ν̂* denotes the corresponding unit vector in direction *ν*. Throughout this paper we set the lattice spacing *a* to *a* = 1, implying that *T* = 1/*N**t* and *L* = *N**s*.\nThe continuum limit for the lattice model is reached via *J* → ∞. The mass gap of the system can be expressed in the bare coupling *J* and a UV cut-off as *m*2 = Λ*U**V*2exp( − 4*π**J*). On the lattice the cut-off is proportional to the inverse lattice spacing, Λ*U**V* = *C*/*a*, and to two loops the mass gap reads,\n \n$$\\begin{aligned}\n am=C(1+2\\pi J)\\exp(-2\\pi J) \\quad\\text{for } J\\to\\infty~.\n \\label{eq\\_two\\_loop\\_massgap}\\end{aligned}$$ \nThe partition sum is defined as the lattice path integral *Z* = ∫D[*r⃗* ] *e*− *S*[*r⃗*], where the measure D[*r⃗* ] is the product over the O(3)-invariant measures for *r⃗*(*x*) on all lattice sites. The particle density and susceptibility are defined as *μ*-derivatives of *Z* as in the continuum,\n \n$$n=\\frac{1}{N\\_s N\\_t}\\,\n \\frac{\\partial \\ln Z}{\\partial \\mu}~,\\qquad\n \\chi\\_n \n = \\frac{\\partial n}{\\partial \\mu}~.\n \\label{eq\\_density\\_susc\\_first}$$ \nAs a check we will also use the expectation value of the action density at *μ* = 0, [2](#fn2)\n \n$$e\n =4-\\frac{2}{N\\_s N\\_t}\\,\n \\frac{\\partial \\ln Z}{\\partial J}~,\n \\label{eq\\_energy\\_first}$$ \nwhose strong and weak coupling expansions are,\n \n$$e =\\begin{cases}\n 4-4y-8y^3-\\frac{48}{5}y^5+\\ldots, & \\text{for small } J,\\\\\n \\frac{2}{J}+\\frac{1}{4J^2}+\\frac{0.156}{J^3}+\\ldots & \\text{for large } J.\n \\end{cases}\n \\label{eq\\_energy\\_expansions}$$ \nwhere $y=\\coth J-\\frac{1}{J}$.\nIn we have introduced the following (exact) representation of the partition function in terms of integer dual variables *m**x*, *ν* ∈ Z and $k\\_{x,\\nu},\\overline{m}\\_{x,\\nu}\\in \\mathbb{N}\\_0$,\n \n$$\\begin{aligned}\nZ= \\label{eq\\_dualZ}\n& \\sum\\_{\\{m,\\overline{m},k\\}} \\left(\\prod\\_{x,\\nu} \\frac{J^{k\\_{x,\\nu}}}{k\\_{x,\\nu}!} \\frac{(J/2)^{|m\\_{x,\\nu}|+\n2\\,\\overline{m}\\_{x,\\nu}}}{(|m\\_{x,\\nu}|+\\overline{m}\\_{x,\\nu})!~\\overline{m}\\_{x,\\nu}!} \\right) e^{\\,\\mu\\sum\\_x m\\_{x,2}} \\\\\n& \\times ~ \\prod\\_x \\mathcal{I}\\left(\\sum\\_\\nu(k\\_{x,\\nu}+k\\_{x-\\hat{\\nu},\\nu}),1+\\sum\\_\\nu\\Big[m\\_{x,\\nu}+\nm\\_{x-\\hat{\\nu}}+2\\,(\\overline{m}\\_{x,\\nu}+\\overline{m}\\_{x-\\hat{\\nu}})\\Big]\\right) \\nonumber \\\\\n& \\times ~ \\prod\\_x \\delta\\left(\\sum\\_\\nu[m\\_{x,\\nu}-m\\_{x-\\hat{\\nu},\\nu}]\\right) ~ E\\left(\\sum\\_\\nu[k\\_{x,\\nu}+k\\_{x-\\hat{\\nu},\\nu}]\\right)~, \\nonumber\\end{aligned}$$ \nwhere the Kronecker delta, the evenness function *E* and a function I (related to the beta function) have been used:\n \n$$\\begin{aligned}\n \\delta(n) \n &=\n\\begin{cases}\n1 &n=0 \\\\\n0 &\\text{else}\n\\end{cases}\n~,\\quad \nE(n) =\n\\begin{cases}\n1 &n~\\text{even} \\\\\n0 &n~\\text{odd}\n\\end{cases}~,\\notag\\\\ \\mathcal{I}(a,b)\n &=\\frac{\\Gamma\\left(\\frac{a+1}{2}\\right)\\,\\Gamma\\left(\\frac{b+1}{2}\\right)}{\\Gamma\\left(\\frac{a+b+2}\n{2}\\right)}~.\\end{aligned}$$ \nNote that the current *m**x*, *ν* is conserved since the Kronecker delta of ∑*ν*[*m**x*, *ν* − *m**x* − *ν̂*, *ν*] ≡ (∇*m*)*x* corresponds to the discrete version of the vanishing divergence condition (at each site *x*). The chemical potential couples to the corresponding sum over the temporal components of *m**x*, *ν*. This sum can be rewritten using the conservation as ∑*x**m**x*, 2 = *N**t*∑*x*1*m**x*, 2 = 1/*T* ⋅ *w*[*m*], where *w*[*m*] is the total winding number of the *m*-loops reflected in the net *m*-flux through every time slice. Thus, the chemical potential appears through weights exp(*μ* ⋅ integer/*T*). This is also the main advantage of dual representations of systems with respect to chemical potentials: if the dual partition function has no sign problem at vanishing *μ* – which holds for our system – *μ* does not introduce a sign problem either.\nIn the dual representation, the observables take the form\n \n$$\\begin{aligned}\n n\n &=\\frac{1}{N\\_s N\\_t}\\big\\langle \\,\\sum\\_x m\\_{x,2}\\big\\rangle\n =\\frac{1}{N\\_s}\\big\\langle w[m]\\big\\rangle~,\n \\label{eq\\_observables\\_dual\\_zero\\_densonly}\\\\\n \\chi\\_n\n &=\\frac{N\\_t}{N\\_s}\\big(\\langle w[m]^2\\rangle-\\langle w[m]\\rangle^2\\big)~,\\label{eq\\_observables\\_dual\\_zero\\_chionly}\\\\\n e\n &=4-\\frac{2}{N\\_s N\\_t}\\,\n\\big\\langle \\,\\sum\\_{x,\\nu}\\big[ \\, k\\_{x,\\nu} + |m|\\_{x,\\nu} + 2 \\, \\overline{m}\\_{x,\\nu} \\, \\big]\\big\\rangle~,\n\\label{eq\\_observables\\_dual\\_one}\\end{aligned}$$ \nwhere here ⟨*O*⟩ is the expectation value of *O* in the dual representation obtained by inserting the expression *O* into the sum in Eq. and dividing by *Z*.\nWe will also measure the space-time average of the third dual variable,\n \n$$K \\; =\\; \\frac{1}{N\\_sN\\_t}\\,\\big\\langle \\,\\sum\\_{x,\\nu}\\, k\\_{x,\\nu}\\big\\rangle~,\n\\label{eq\\_thirddual}$$ \nas a measure for the anisotropy of the system. For its interpretation we sketch how the dual representation is obtained for the third variable, using that $Z\\propto\\int \\mathcal{D}[\\vec{r}\\,] \\prod\\_{x,\\nu}\\sum\\_{k\\_{x,\\nu}=0}^\\infty \\big[J\\,r\\_3(x)r\\_3(x+\n\\hat{\\nu})\\big]^{k\\_{x,\\nu}}/k\\_{x,\\nu}!$. Evaluating ([eqthirddual]) inserts another *k**x*, *ν* into the dual partition sum and reduces the argument of the factorial by one, which can be compensated by a shift of the summation variable giving a factor of *J* and the hopping term. It follows that ⟨*k**x*, *ν*⟩ = *J* ⟨*r*3(*x*)*r*3(*x* + *ν̂*)⟩, where the latter expectation value is in the conventional representation. Hence *K* measures the amount of hopping in the direction perpendicular to the *x* − *y* plane where we excite O(2) angular momentum. Thus, we expect *K* to be small in the anisotropic phase at large *μ*.\nAt vanishing *μ* the theory enjoys the full O(3) symmetry and the amount of hopping must be the same for all components. In the energy density above this can be seen by virtue of the fact that $|m|\\_{x,\\nu} + 2 \\, \\overline{m}\\_{x,\\nu}$ is the sum of two dual variables of the same nature as *k**x*, *ν*; see Eq. (12) of. Therefore, *e* = 4 − 6*K*/*J* should hold at *μ* = 0.\nIn the same way the chemical potential couples to the integrated temporal component of the conserved O(2) current, the twist-induced imaginary spatial chemical potential *i**φ*/*L* couples to the integrated spatial component. The partition function in the presence of the twist thus has an additional factor exp(*i**φ*/*N**s* ⋅ ∑*x**m**x*, 1) = exp(*i**φ* *w**s*[*m*]), where *w**s*[*m*] is the total spatial winding number of *m* in each configuration. In the definition of the spin stiffness *σ*, Eq. , the second derivative with respect to *φ* brings down − *w**s*[*m*]2 and setting *φ* to zero afterwards results in the expectation value one obtains\n \n$$\\begin{aligned}\n {\\sigma}=\\frac{N\\_s}{N\\_t}\\, \\langle w\\_s[m]^2\\rangle\n =LT\\, \\langle w\\_s[m]^2\\rangle\\;,\\end{aligned}$$ \nin the dual repesentation without twist (similar to *χ**n* in Eq. ).\nNumerical simulation, tests and basic analysis\n==============================================\nIn this section we collect several more technical aspects of this paper. We briefly discuss our simulation strategy for the dual formulation and evaluate the correctness of its results by comparing them to perturbative strong and weak coupling calculations. Furthermore we present numerical results for the phase diagram at finite volume and temperature as well as a finite volume scaling analysis which indicates that at nonzero temperature all transitions are smooth crossovers.\n![Energy density versus the coupling at \\mu=0. The analytical results for the strong and weak coupling expansions from Eq. agree very well with the numerical results (having very small error bars) obtained from simulating the dual ensemble on a 10\\times10 lattice and using Eq. .](Plots/o3_2dim_dual_worm_energy_strong_weak_vs_j_ \"fig:\") [figenergy]\nDual simulation and tests\n-------------------------\nIn this subsection we briefly discuss Monte Carlo simulation strategies for the dual representation Eq. ([eqdualZ]) and present tests for its correctness. Obviously each term in the partition sum ([eqdualZ]) is real and positive and a probability interpretation of the weights of the dual configurations is possible. The remaining challenge of a dual Monte Carlo simulation is to generate only those configurations that obey all the constraints. For the unconstrained dual variables $\\overline{m}\\_{x,\\nu}$ conventional local Metropolis updates are sufficient. For the constrained variables *m**x*, *ν* and *k**x*, *ν* the constraints enforce closed loops and loops where flux is conserved modulo 2, respectively. In both cases one can generate new admissible configurations by changing the variables along an arbitrarily chosen closed loop which guarantees that the constraints remain intact. This loop along which one updates the dual variables can for example be grown in steps using local random choices and corresponding Metropolis decisions until it closes, which is the well known worm strategy – this is the update used here. After equilibration, we typically use 105 to 106 measurements on configurations separated by ${\\mathcal O}(20)$ sweeps. The statistical errors we show are determined with the jackknife method taking autocorrelation times into account.\n\n![Top: Expectation value of the particle number density versus the chemical potential \\mu, both in lattice units, at different couplings J for lattice size 90 \\times 90. Bottom: Logarithm of the corresponding critical chemical potentials as a function of the coupling. We compare our data to the strong coupling result \\ln(3/J) and the weak coupling expansion of the mass gap and Eq. , having obtained C=102(2) from a fit of the J\\geq 1.4 data.](Plots/o3_2dim_dual_worm_mcrit_vs_j_ \"fig:\") [figpartvsmufirst]\nFor a first test we computed the energy density at vanishing chemical potential, cf. Eq. . In Fig. [figenergy] we show the results for *e* at *μ* = 0 as a function of the coupling *J* and compare to the weak and strong coupling expansions from Eq. . We find excellent agreement in the corresponding domains of *J*. This demonstrates that the mapping to the dual variables and the implementation of the dual Monte Carlo simulation are correct.\nWe now switch to the situation where the dual approach is really essential, i.e., the simulations at nonzero chemical potential. In the top panel of Fig. [figpartvsmufirst] we plot the particle density *n* measured via Eq. as a function of *μ*. As a general phenomenon at low temperatures a net density is induced into the system only after *μ* has reached a *threshold*, the mass of the lightest particle with charge coupling to *μ*. The critical values of *μ* visible in that plot thus depend on the coupling *J* just like the mass. This is shown in the bottom panel of Fig. [figpartvsmufirst], again with strong and weak coupling expansions. The former, *μ* = ln(3/*J*) for *J* → 0, is worked out in App. [appstrongcouplingmus], whereas the latter follows from the two-loop mass gap formula, Eq. , in which we obtain the constant *C* related to the UV cut-off by a one-parameter fit. The agreement is again very good, which is seen also in a comparison to the literature (Fig. 10 of ). Note also that the continuum scaling sets in at *J* ≃ 1.4, as for the energy density in Fig. [figenergy].\nMost of our lattice data were taken at *J* = 1.3, where *a**m* ∼ 0.22. This means a restriction to *μ* ≪ 5*m*, otherwise *a**μ* becomes comparable to 1 and strong discretization effects set in.\nFinite lattice phase diagram\n----------------------------\n![Phase transition in the J-\\mu plane for N_s=30 and N_t=1000. At large J, i.e., towards the continuum limit, the critical chemical potentials with increasing index (from red to green to black to blue) induce individual transitions between integer charges as utilized in. At small J the first N_s of them join to a single transition, \\mu_1=\\ldots=\\mu_{Ns}, increasing the particle density n in lattice units from 0 to 1 (see text). The next critical chemical potential \\mu_{N_s+1} (blue) is separated from those, and above it a density n=2 will be induced. At large J, on the other hand, the change from \\mu_{Ns} to \\mu_{N_s+1} plays no particular role. The strong coupling predictions for the transitions 0\\to 1 and 1\\to 2 are included as solid curves.](Plots/o3_2dim_dual_ns30_nt1000_mucritn_vs_j \"fig:\") [figmanymucs]\nBefore we come to discussing continuum results, we first consider nonzero lattice spacing and finite volume, i.e., we study the system at fixed lattice size without a final continuum limit *J* → ∞. We determine the *J*-*μ* phase diagram at low temperature, using the onset of non-zero particle density as a function of *μ* for determining the phase boundaries. Fig. [figmanymucs] shows this phase diagram for several critical chemical potentials. Note that the number of temporal sites is fixed, so the temperature in mass units *T*/*m* = 1/(*N**t**a**m*) varies with the coupling according to the mass gap formula *a**m*(*J*), e.g., Eq. .\nIn we have shown that close to the continuum limit at small temperatures and volumes, the particles are induced into the system one by one displaying integer plateaus in the particle number itself (not its density). These transitions show up for large *J* and some examples are plotted in Fig. [figmanymucs]. The locations of these critical *μ*’s are governed by the particle interaction phase shifts and thus contain interesting physics, as explained in detail in.\nIn the strong coupling regime at small *J* the situation is different. In the dual representation of the partition function, Eq. , every dual variable is suppressed by the corresponding power of *J*. Still, the temporal components of the flux variable *m* are promoted by factors of *e**μ*, that eventually overcome the factors of *J*/2. This mechanism acts locally on every spatial site, which means that if it is preferable to have *Q* particles at some site (*m**x*, 2 = *Q*), then this immediately applies to all sites. In fact, the superposition of *Q* fluxes does not cost any action at strong coupling. Therefore, in this regime the particle number *density* changes by one (in lattice units). The corresponding critical values of *μ* depend on *J* logarithmically, and their values are computed in Appendix [appstrongcouplingmus]. Fig. [figmanymucs] shows good agreement with these curves for small *J* and illustrates how intermediate couplings interpolate the transitions between these regimes. In particular all critical values of *μ* are on equal footing towards the continuum (large *J*), whereas in the strong coupling regime (small *J*) multiples of *N**s* bunch, such that regimes with fixed lattice density open up.\n![Phase transition as in Fig. [figmanymucs], but with a modified quantity on the y-axis (and some more data points for higher J’s), see text.](Plots/o3_2dim_dual_ns30_nt1000_mucritn_vs_j_final \"fig:\") [figmanymucso2]\nIn Fig. [figmanymucso2] we show the lattice phase diagram with the quantity *J*/2 ⋅ exp(*μ**Q*) on the *y*-axis. In the strong coupling limit *N**s* critical *μ*’s now meet at half integers 3/2, 5/2 etc., which agrees with the derivation in Appendix [appstrongcouplingmus] again. The corresponding diagram for the O(2) model is shown in Fig. 3 of Ref. (for this model the curves meet at integer values).\nLet us conclude this subsection with a quick look at the third dual variable *k**x*, *ν*, or more specifically at its sum *K* as defined in Eq. . For *μ* larger than the threshold the O(3) symmetry is explicitly broken to O(2) by the presence of the O(2) charge, which is expected to be manifest in a decrease of *K* (see the discussion below Eq. ). In Fig. [figthirddual] we show *K* as a function of *μ* (normalized by *μ**c*) and indeed find the onset of a drop at the critical chemical potential, which confirms the picture that the system tends to become more and more planar as *μ* increases.\n [figthirddual]\nCrossover at nonzero temperature\n--------------------------------\nWe conclude the first analysis of the lattice model by studying the nature of the transitions mapped out in the previous subsection. For this study we still keep the temperature *T* fixed and perform the standard scaling analysis of order parameters with the spatial volume *L* to analyze the nature of the transition.\n [figvolumescaling]\nFig. [figvolumescaling] shows the particle number density (top panel) and its susceptibility (bottom panel) at a fixed low temperature of 2% of the mass. As a function of chemical potential *μ* the particle number *n* is monotonically increasing, with a strong variation emerging only above *μ**c*, while the susceptibility displays a maximum. Doubling the spatial size, the data still fall on top of each other for both observables. Equivalent results were found for other nonzero temperatures and we conclude that for nonzero temperature the transition is smooth, i.e., it is a *crossover*.\nQuantum phase transition\n========================\nWe now explore the possibility of a quantum phase transition at zero temperature, i.e., we focus on the combined limit of zero temperature and infinite volume, i.e., *T* = 0, *L* = ∞ and analyze the transitions of the O(3) model as a function of the chemical potential. This amounts to sending the extents of both, Euclidean time and space to infinity. Of course, with finite numerical simulations this can only be done as a limit, *T* → 0 and *L*− 1 → 0. However, in general, the outcome depends on the particular choice of the trajectory towards that limit which one chooses in the *T*-*L*− 1 plane. In particular the spatial and temporal correlation lengths can be different, as expressed by the dynamical critical exponent (see Sec. [secqptstiff] below). Thus we will discuss the behavior of the system for different “scaling trajectories”, i.e., different paths leading to the limit *T* = 0, *L*− 1 = 0. For these scaling trajectories we will present and interpret our data for the particle density and the spin stiffness. For the interpretation we will partly rely on simple model systems showing characteristic features observed in our simulations.\nScaling trajectories towards zero temperature and infinite volume and particle number results\n---------------------------------------------------------------------------------------------\nAs already outlined, when considering the limits *T* → 0 and *L*− 1 → 0, the behavior of the system will depend on the particular scaling trajectory one follows in the *T*-*L*− 1 plane towards the origin. These limits can be taken in different ways, and a possible choice is to take the *L*− 1 → 0 limit first, keeping temperature fixed, and then take *T* → 0. We can also keep *L* fixed and take *T* → 0 first, and then *L*− 1 → 0. More generally we can take the limit *T*, *L*− 1 → 0 keeping [eq:scaling] TL= fixed, where *α* is a nonnegative real number. Generically, the constant has a noninteger mass dimension, for the practical implementation (e.g., for *α* = 2 below) one can fix (*T*/*m*)(*L**m*)*α* to a dimensionless constant. Notice that the consecutive limits mentioned above correspond to the limiting cases *α* = 0 and *α* = ∞. Here we will consider these two scaling trajectories plus *α* = 1, 2 as we illustrate in Fig. [figscalingdiagram]. In this subsection we briefly discuss these different scaling trajectories and partly describe the resulting physics using model calculations for the behavior at the emerging phase transitions.\n(0,0) circle (0.15); (0,0.3) – (0,4) – (3.2,4); (0,4.2) – (0,5); (0.3,0) – (4,0) – (4,3.2); (4.2,0) – (5,0); (0.3,0.3) – (3.4,3.4); (0.3,0.05).. controls (2.2,0.25) and (3,0.5).. (3.85,3.2); (T) at (-0.4,4.8) *T*; (L) at (5.5,0) *L*− 1; (a0) at (1.9,4.3) *α* = 0; (a1) at (1.4,2.4) *α* = 1; (a2) at (2.35,1.43) *α* = 2; (ainf) at (4.9,1.9) *α* = ∞;\n[figscalingdiagram]\nIn the limiting trajectories *α* = 0 and *α* = ∞ one has to perform the two limits consecutively. This is difficult to implement on the lattice but reveals interesting physics. Therefore, we will use simple model calculations to illustrate characteristic features of the consecutive limits. We will compare our numerical data to results for *free one-dimensional fermions*: At low densities the behavior is governed by the small momentum-exchange between the particles, which for the O(3) model at hand are one-dimensional repulsive bosons. At low momentum their behavior is universal and given by free (spinless) fermions, see Sec. [seccomparison].\nFor the other two scaling trajectories, *α* = 1 and *α* = 2 the results will not differ much concerning the particle density, but are characterized by a different behavior of the spin stiffness, which we will discuss in detail in Subsection [secqptstiff].\n \n \nThe *α* = 0 trajectory corresponds to the consecutive limits \n*L*− 1 → 0at fixed *T*, then *T* → 0 . \n The first limit of this sequence corresponds to the finite volume scaling at fixed nonzero temperature studied in Subsection [sectionfinitevolume]. This analysis has revealed a *crossover* as seen from Fig. [figvolumescaling]. Actually every fixed temperature possesses a specific crossover curve *n*(*μ*)*T*, which upon lowering the temperature *T* to zero in the second limit might become steeper near some *μ**c* and turn into a genuine phase transition. To some extent this feature is seen in our data shown in Fig. [figtempscaling] below, where it is, however, overlaid by the formation of “condensation steps” (see the discussion below).\n![Emergence of a phase transition in the limit T \\to 0 for free one-dimensional fermions in an infinite volume. We show that the smooth curves for T/m=1/10 (green) and T/m = 1/50 (blue) approach the T = 0 square root behavior from Eq. . This is a model calculation for the scaling trajectory \\alpha=0, especially its second limit T\\to 0 at L^{-1}=0.](Plots/dens_ferm_infL_Ttozero \"fig:\") [figfermroot]\nThe emergence of such a quantum phase transition as *T* approaches 0 is of course a well known feature which already appears in the simple model of free fermions in one spatial dimension: The density *n* is given by the Fermi-Dirac integral (*L*− 1 = 0),\n \n$$\\begin{aligned}\n n\\_f = \\int\\_{-\\infty}^\\infty\\!\\frac{dk}{2\\pi}\\,\\frac{1}{1+\\exp\\big((m+k^2/2m-\\mu)/T\\big)} \\;,\n \\label{eq\\_ferm\\_one}\\end{aligned}$$ \nwhere we use a nonrelativistic dispersion which is sufficient for a first illustration. The density can be expressed by a polylogarithm Li1/2, from which one obtains a square root in the *T* → 0 limit,\n \n$$\\begin{aligned}\n \\frac{n\\_f}{m} \\to \n &\\:\\frac{\\sqrt{2}}{\\pi}\\sqrt{\\mu/m-1}\n \\cdot\\Theta\\_{\\text{Heaviside}}(\\mu/m-1)\\notag\\\\\n &\\:\\mbox{for} \\quad L^{-1}=0,\\, T \\to 0 \\;.\n \\label{eq\\_ferm\\_two}\\end{aligned}$$ \nThe square root behavior, which is non-analytic at *μ*/*m* = 1, is indeed reached only for *T* = 0 via a sequence of analytic crossover-type curves as shown in Fig. [figfermroot].\n \n \nWe have already pointed out that our results for the density as a function of *μ* start to show “condensation steps” when the temperature *T* is lowered at fixed *L*. Thus it is interesting to study also the scaling trajectory *α* = ∞, which is defined by the consecutive limits\n \n*T* → 0at fixed *L*, then *L*− 1 → 0 . \nOur numerical results for the first one of these two limits are shown in Fig. [figtempscaling]. When lowering the temperature *T* at fixed *L* we observe the emergence of plateaus in the density where the particle numbers are integers. The plateaus correspond to sectors with fixed particle number, and are smoothed out by the finite temperature (the width of the transition region being proportional to the temperature). Therefore, sharp *steps* emerge in the *T* → 0 limit at any fixed *L*.\n [figtempscaling]\nAfter considering *T* = 0 at finite *L* for following the *α* = ∞ trajectory we need to perform the second limit *L*− 1 → 0. Also the behavior of this second limit can be illustrated in a one-dimensional fermion model, the Tonks-Girardeau limit of the Lieb-Liniger system, which we discuss in detail in Section [seccomparison]. One finds that the minimal energy in each particle number sector *Q* is *E*min(*Q*) = *π*2/(6*L*2*m*) ⋅ (*Q*2 − 1)*Q*. Steps occur whenever *E*min(*Q*) − *Q**μ* = *E*min(*Q* − 1) − (*Q* − 1)*μ* and thus are located at *μ**Q* = *m* + *π*2/2 ⋅ *Q*(*Q* − 1)/(*L*2*m*). The corresponding density *n*/*m* = 1/*L**m* ⋅ ∑*Q* = 1∞ΘHeaviside(*μ* − *μ**Q*) is shown in the top panel of Fig. [figstepslargervolumes]. It ‘oscillates’ around its square root limit reaching it by ever smaller oscillations. The square root can also be seen analytically, since for large *Q* the steps are at *μ**Q*/*m* = 1 + *π*2/2 ⋅ (*n*/*m*)2.\n\n [figstepslargervolumes]\nThe same behavior is seen in our numerical O(3) data shown in the bottom panel of Fig. [figstepslargervolumes], where, in the same fashion, ‘oscillations’ around a limiting curve diminish with increasing *L*. However, this can only be done at low but nonzero temperatures and, as discussed above, this gives rise to a crossover instead of a phase transition.\nIn the scaling trajectories *α* = 1 and *α* = 2 we consider approaches to the limit *T* = 0, *L*− 1 = 0 by sending *T* and *L*− 1 to zero simultaneously. The trajectories differ in their functional relation between *T* and *L*:\n \n$$\\begin{aligned}\n \\alpha=1: \\quad \n &T\\to 0 \\text{ and }L^{-1} \\to 0\\; \\text{ with }T = L^{-1}\\\\\n \\alpha=2: \\quad \n &T\\to 0 \\text{ and }L^{-1} \\to 0\\; \\text{ with }T \\propto (L^{-1})^2\\end{aligned}$$ \n [figdensitiesbothscalings]\nObviously *α* = 1 corresponds to square lattices *N**t* = *N**s*, while *α* = 2 corresponds to time elongated lattices with *N**t* ∝ (*N**s*)2. To be specific, in our numerical simulations for *α* = 2 we have used *N**t* = (*N**s*/2)2 at *J* = 1.3 (*a**m* ∼ 0.22), which amounts to *T*/*m* = 0.88/(*L**m*)2 or $Lm=0.94/\\sqrt{T/m}$.\nFig. [figdensitiesbothscalings] shows our simulation results for these two scaling trajectories. We show the particle density *n* as a function of *μ* for *T* → 0 and *L* → ∞ simultaneously. One can see in both trajectories that *n* vanishes for values of *μ* smaller than the threshold *m*, while it approaches certain nonzero values for *μ* > *m*. This is an indication for a nonanalyticity developing at *μ* = *m* in the limit *T* → 0, and thus for a *quantum phase transition*. Whether it emerges in the form of a jump in *n* (first order transition) or an infinite slope (second order) cannot be decided with our current data (and the common infinite volume limit *L* → ∞ to distinguish the orders is part of the scaling to zero temperature). The agreement with analytic predictions for repulsive bosons presented in Sec. [seccomparison] indicates a square root dependence on *μ* and thus a second order transition.\nMoreover, the data at lowest *T*/highest *L* do not differ much between *α* = 1 and *α* = 2. This bulk quantity therefore does not seem too sensitive to the particular scaling trajectory in the limit *T* → 0 and *L* → ∞.\nSpin stiffness and dynamical critical exponent\n----------------------------------------------\n [figstiffnessbothscalings]\nFig. [figstiffnessbothscalings] shows our spin stiffness data for the two scaling trajectories *α* = 1 and *α* = 2. Qualitatively, the stiffness behaves like the density: it vanishes for *μ* < *m* and is nonzero for *μ* > *m*. In contrast to the density, however, the stiffness shows a clear dependence on *α*: for *α* = 1 it is *significantly smaller* in the *μ* > *m* phase than for *α* = 2. In addition, the *stiffness agrees with the density* (divided by mass) for *α* = 2 at small densities, as Fig. [figstiffnessvsdensity] shows. Our stiffness data do not display the signature of a BKT transition for *μ* > *m* in the spatial correlations suggested in.\n [figstiffnessvsdensity]\nLet us now come to the interpretation of these data. It is known that the stiffness depends on the order of the limits *T* → 0 and *L* → ∞ : when *L* → ∞ is taken first (i.e. *α* = 0 scaling) the stiffness vanishes, whereas when *T* → 0 is taken first (i.e. *α* = ∞ scaling) it approaches the susceptibility of the ground state with respect to the same twist.\nFor *α* = 0 one keeps the temperature fixed (and thus the transition is a crossover), and then increases the spatial size to infinity. It is clear that in this way one looses all spatial correlations, the twist at the spatial boundary has no effect on the free energy for *L* → ∞ and the stiffness vanishes (the successive limit of zero temperature cannot change this any more). Our data for *α* = 1 suggest that this scaling trajectory is similar to *α* = 0 since the system size is growing too fast in comparison to *T* → 0, such that spatial correlations are lost leading to a *small stiffness*.\nFor *α* = ∞ one considers the zero temperature system at fixed volume and only afterwards the size is increased. At zero temperature one expects the ground state to determine the thermodynamic properties. In fact, the system at finite *L* is gapped proportional to 1/*L*2. Therefore, the partition function in the *T* → 0 limit is dominated by the ground state, and the free energy becomes the ground state energy (noting that the latter is *T*-independent, this is nothing but the third law of thermodynamics). As a consequence, the stiffness is the susceptibility of the ground state with respect to the spatial twist, as mentioned above.\nOn top of that, we now invoke an argument from the Bethe ansätze of Sec. [seccomparison]. One can easily convince oneself [3](#fn3) that the (nonrelativistic) pseudo-momenta *k**i* in the presence of a spatial twist *φ* are shifted by *φ*/*L*. The ground state energy ∑*i* = 1*Q**k**i*2/(2*m*) receives no linear term in *φ* since the total momentum ∑*k**i* vanishes. The quadratic term is the constant *Q* ⋅ (*φ*/*L*)2/(2*m*), and we immediately obtain the stiffness as *σ* = *Q*/(*L**m*) = *n*/*m*. This explains that the *stiffness equals the density* for *α* = ∞, for small densities. Our data for *α* = 2 suggest that this scaling trajectory is similar to *α* = ∞ since the stiffness still equals the density, as seen in Fig. [figstiffnessvsdensity].\nFinally, let us discuss the *dynamical critical exponent* *z* that results from our findings. The correlation length near a second order phase transition is known to diverge. To achieve *T* = 0 in the quantum phase transition one has to send the extent of the Euclidean time *β* to infinity (just what we performed in the lattice simulations), and only at *β* = ∞ the phase transition occurs and the spatial correlation length *ξ* is infinite. However, the system also has a typical correlation length *ξ**τ* in the Euclidean time direction. If nothing in the system distinguishes between space and Euclidean time, the two correlation lengths have to agree (which corresponds to *z* = 1 below). However, in general – and in particular in the presence of *μ* which breaks the Euclidean symmetry by coupling to temporal components/winding numbers – the two correlation lengths are related by *ξ**τ* ∼ *ξ**z* with *z* the dynamical critical exponent. Note the similarity to our scaling trajectories *β* ∼ *L**α* and indeed our stiffness data can be used to determine *z* and the correlation length critical exponent *ν*. For that we use finite size scaling of the free energy density,\n \n$$f\\sim\\frac{1}{L\\beta}\\,g\\big(\\frac{\\xi}{L},\\frac{\\xi^z}{\\beta};\\varphi\\big)$$ \nwhere the prefactor is the inverse volume (in 1 + 1 dimensions) making *f* = *F*/*L* an intensive quantity and *g* is a universal function of the spatial correlation length, the box sizes and the twist angle. Again, noninteger dimensions can be compensated by the corresponding powers of the mass. Using the scaling *ξ* ∼ *δ*− *ν* with the normalised distance to the critical value *δ* = *μ*/*m* − 1, and the definition we get for the stiffness\n \n*σ* ∼ *L*1 − *z* *h*(*L*1/*ν**δ*, *T**L**z*) \nwhere *h* is another universal function.\n [fig:finitesizefirst]\nFor practical purposes one can derive two criteria :\n \n$$\\begin{aligned}\n (a) &\\text{ the curves }\\mu\\text{ vs.\\ }L^{z-1}{\\sigma}\\text{ intersect at }\\mu=m,\\\\\n &\\text{ if }TL^z\\text{ is kept constant}\\end{aligned}$$ \nsimply because at *δ* = 0 the length *L* does not enter as argument of *h* anymore and\n \n$$\\begin{aligned}\n (b) &\\text{ the curves }L^{1/\\nu}\\delta\\text{ vs.\\ }L^{z-1}{\\sigma}\\text{ collapse to a single curve,}\\\\ \n &\\text{ again if }TL^z\\text{ is kept constant}\\end{aligned}$$ \nOur data are collected at constant *T**L**α* with *α* = 1, 2 and thus we can ask whether *z* = 1 or *z* = 2 obey these statements, at least approximately. We will rescale all dimensionful quantites by the corresponding power of mass.\nFor statement (a) with *z* = 1 we thus plot *σ*(*μ*) keeping *T**L* constant, i.e., on the scaling trajectory *α* = 1, while for *z* = 2 we plot *L*1*σ*(*μ*) keeping *T**L*2 constant, i.e., on the scaling trajectory *α* = 2, and see whether the data for different *L*’s intersect. As Fig. [fig:finitesizefirst] shows, the curves from our numerical data intersect approximately for *z* = 2 (albeit at *μ* slightly smaller than *m*), whereas for *z* = 1 they certainly do not intersect.\n [fig:finitesizesecond]\n [fig:finitesizethird]\nFor statement (b) one has to deal with the appearance of another critical exponent *ν*. In Fig. [fig:finitesizesecond] we plot the obervables of this statement assuming the value *ν* = 1/2 for free fermions. Again a collapse to a single describes the situation much better for *z* = 2. Finally, we can assume *z* = 2 and check the value of *ν* away from 1/2. Fig. [fig:finitesizethird] shows data proposing *ν* = 1/3 and *ν* = 3/4, for which the curves certainly do not collapse.\nFrom this analysis we conclude that the critical exponents of the O(3) quantum phase transition at *μ* = *m* are consistent with *z* = 2 and *ν* = 1/2, the values of free 1d fermions.\nComparison to analytic results on repulsive bosons\n==================================================\nIn this section we will compare our density data to the density for systems of repulsive bosons in one spatial dimension. One of the best known examples is the one introduced by Lieb and Liniger (LL), where boson pairs interact via a Dirac delta function. The system can be solved in terms of plane waves picking up phase shifts *δ* when two bosons are interchanged. Such a Bethe ansatz works for nonrelativistic or relativistic bosons, without antibosons though.\nThe phase shifts of the O(3) model are also known and those of “isospin 2”,\n \n$$\\begin{aligned}\n \\delta=-\\arctan\\frac{\\pi}{2\\theta}~,\\end{aligned}$$ \nwith the relative rapidity,\n \n$$\\begin{aligned}\n \\theta=\\text{arsinh}(k/m) \\;,\\end{aligned}$$ \ngovern the low density regime. We will thus use the Bethe ansatz equations with those phase shifts. Note that the O(3) wave functions are not plane waves anymore, but the Bethe ansatz is believed to give an exact result at low densities.\nAt very low momenta the O(3) phase shifts agree with those of the LL model, actually all repulsive one-dimensional bosons are universal in the deep IR, where the precise UV shape of the interaction is not relevant and where *δ*(0) = − *π*/2 by Levinson’s theorem. Using only this value one arrives at the Tonks-Girardeau (TG) gas, which is the infinite coupling limit of the LL model. Its eigenvalues are known to be that of free fermions with the modification that the numbers *I**i* in Eq. below are not always half-integers as is the case for antiperiodic fermions (which is not relevant for large *L*) and that the eigenfunctions of the TG gas are still symmetric under exchange of bosons. We have already shown that this results in the square root behavior of the particle density for *μ* ≈ *m* at zero temperature and infinite volume (see also Eq. below).\nTo be more precise, the relativistic Bethe ansatz reads\n \n$$\\begin{aligned}\n Lm \\sinh(\\theta\\_i) -2\\sum\\_{\\substack{j=1\\\\j\\neq i}}^Q\\Delta(\\theta\\_j-\\theta\\_i)=2\\pi I\\_i~,\n \\label{eq\\_bethe\\_ansatz}\\end{aligned}$$ \nwhere the *I**i* are distinct, and half-integer/integer for even/odd total charge *Q*. In our conventions the dynamics enters via the phase shifts in [eq:Delta] =+/2 with lim*k* → 0Δ = 0. This ansatz works for ground states (with the choice {*I**i*} = { − (*Q* − 1)/2, …, (*Q* − 1)/2}) and excited states at fixed particle number *Q* in any finite volume *L*. The nonrelativistic Bethe ansatz can be obtained by approximating the rapiditites as *θ* ≈ *k*/*m*. Both are not too hard to solve numerically.\n \n [fig:latticecompare]\nFurthermore, Yang and Yang have derived a thermodynamic Bethe ansatz for the density *ρ*(*k*) and energy density *ε*(*k*) at nonzero temperatures and chemical potentials in infinite volumes. The non-relativistic Yang-Yang equations give rise to\n[eq:YYnonrel]\n \n$$\\begin{aligned}\n 2\\pi\\rho(k)[1+e^{\\epsilon(k)/T}]\n &=1+\\int\\_{-\\infty}^\\infty\\!d\\tilde{k}\\,\n \\rho(\\tilde{k})\\,\\Delta'(k-\\tilde{k})~,\\\\\n \\epsilon(k)\n &=\\frac{k^2}{2m}+(m-\\mu)-\\frac{T}{\\pi}\\\\\n &\\times\\!\\!\\int\\_{-\\infty}^\\infty\\!\\!\\!d\\tilde{k}\\,\n \\log\\big(1+e^{-\\epsilon(k)/T}\\big)\\,\n \\Delta'(k-\\tilde{k})\\,.\\notag\\end{aligned}$$ \nwhile, following similar arguments of, and ignoring thermal pair production[4](#fn4), the relativistic Yang-Yang equations read\n[eq:YYrel]\n \n$$\\begin{aligned}\n 2\\pi\\rho(\\theta)\\big(1+e^{\\epsilon(\\theta)/T}\\big)\n &=m\\cosh\\theta+2\\int\\_{-\\Theta}^{\\Theta}d\\tilde\\theta\\;\\rho(\\theta)\\Delta'(\\theta-\\tilde\\theta)\\;,\\\\\n \\epsilon(\\theta)\n &=m\\cosh\\theta-\\mu\n -\\frac{T}{\\pi}\\\\\n &\\times\\!\\int\\_{-\\Theta}^{\\Theta}\\!\\!d\\tilde\\theta\\;\\log(1+e^{-\\epsilon/T})\\Delta'(\\tilde\\theta) \\;.\\notag\n \\end{aligned}$$ \nIn both relativistic and non-relativistic equations, $\\Delta'(x)=\\frac{d\\Delta(x)}{dx}$, where Δ is defined by. Having solved this system one can extract the particle density as *n* = ∫− ∞∞ *d**k̃* *ρ*(*k̃*).\nFinally, the zero temperature limit for the density is,\n \n$$\\begin{aligned}\n 2\\pi\\rho(k)\n &=1+\\int\\_{-K}^K\\!d\\tilde{k}\\,\n \\rho(\\tilde{k})\\,\\Delta'(k-\\tilde{k})~,\\quad \\rho(|k|>K)=0~,\\end{aligned}$$ \nwhere *K* is the analogue of the Fermi momentum related to *μ* by $K=\\sqrt{2}\\,m\\sqrt{\\mu/m-1}$. Low densities amount to *μ* ≈ *m* and thus *K* → 0, for which the integration range shrinks, *ρ*(0) = 1/(2*π*) and *n* = *ρ*(0) 2*K* such that\n \n$$\\begin{aligned}\n \\frac{n}{m}\n =\\frac{\\sqrt{2}}{\\pi}\\, \\sqrt{\\mu/m-1}\n \\qquad(T=0,\\text{ small }n/m)~,\n \\label{eq\\_square\\_root\\_again}\\end{aligned}$$ \nconfirming once more the square root behavior for low densities.\nWe solve these integral equations numerically and compare to the lattice data in Fig. [fig:latticecompare]. The agreement is fairly good. Note that these are parameter-free ansätze, so no fit is involved in the comparison. Let us repeat that the agreement is expected for low temperatures only, since Bethe ansätze do not contain antibosons and pair production.\nSummary\n=======\nIn this paper we have presented thermodynamic lattice simulations of the asymptotically free two-dimensional O(3) model. The complex action problem of the conventional representation at finite chemical potential is overcome by using a representation in terms of dual variables, i.e., we simulated worldlines. In the O(3) model the mass of the particle triplet is generated dynamically, and in our simulations we have confirmed the expectation that at low temperatures a nonzero particle number (or charge) density occurs only when *μ* reaches the threshold given by the mass. Furthermore, in small volumes several critical values of *μ* occur which correspond to integer particle numbers induced by *μ* (for finite lattice couplings see the lattice phase diagram in Sec. [seclatticediagram]).\nA finite volume scaling analysis has revealed that at nonzero temperatures the particle density as a function of *μ* is regular, i.e., the transition is a crossover. The possibility of a phase transition at zero temperature has been analyzed through a simultaneous scaling of Euclidean time and space. Indeed, the data indicate a quantum phase transition at *μ* = *m*, which, using the agreement with Bethe ansätze for one-dimensional repulsive bosons, should be of second order because the particle density follows the universal square root behavior. Further comparisons to these ansätze, including a nonzero temperature, have shown that the O(3) model can indeed be described by these bosons. A more detailed analysis of this finding should include the continuum limit of our lattice data (which we believe will not change the results significantly, as we have observed for some observables at lattice coupling *J* = 1.4), but is beyond the scope of the present study.\nWe have also measured the spin stiffness which in the conventional representation is defined via twisted spatial boundary conditions and in the dual representation measures spatial winding numbers of the worldlines. Based on the boson description at zero temperatures with only ground states, the stiffness is expected to be equal to the particle density. This is consistent with our low temperature data, when scaling *L* such that *T**L*2 is constant, i.e., *α* = 2. However, when *L* is larger in the zero temperature limit, e.g., when *T**L* is kept constant, i.e., *α* = 1, then this equality does not hold and the spin stiffness is significantly lower, which indicates a lost correlation between the spatial boundaries. From these findings we conclude that the dynamical critical exponent *z* is close to 2, in agreement with free fermions.\nAlthough originally introduced for studying a potential BKT transition, our stiffness data have not indicated such a transition in the O(3) system at chemical potentials larger than the mass where it tends to be planar (as we have also confirmed).\nBesides higher precision, further studies in various directions would be useful: First of all, perturbation theory should match our lattice data at high *μ*’s (keeping *a**μ* small to avoid discretization effects). Secondly, measuring the vortex correlation functions or the topological susceptibility should shed light on the possibility of a BKT-like transition. Extensions to higher O(N) or CP(N-1) models as well as to 2+1 dimensions could be done straightforwardly with the dual representation.\nThe physically very interesting regimes of complex chemical potentials, where fractional instantons should occur (which in turn underly the resurgence program mentioned in the introduction), and at large theta angle, with the Haldane conjecture of a phase transition at theta angle equal to *π*, remain a challenge as the dual variables used here do not solve the sign problem in these situations.\nWe thank Quirin Hummel, Yannick Meurice, Lode Pollet, Klaus Richter, Juan Diego Urbina, Andreas Wipf and in particular Hans-Gerd Evertz for helpful discussions. FB is supported by the DFG (BR 2872/6-1) and TK by the Austrian Science Fund, FWF, DK *Hadrons in Vacuum, Nuclei, and Stars* (FWF DK W1203-N16). Furthermore this work is partly supported by the Austrian Science Fund FWF Grant. Nr. I 1452-N27 and by the DFG TR55, *”Hadron Properties from Lattice QCD”*. TS acknowledges the support of the DOE grant E-SC0013036.\nCritical *μ*’s at strong coupling\n=================================\nIn this appendix we derive the critical values of the chemical potential in the strong coupling limit which we use for comparing to our numerical data in Figs. 2 and 3.\nFor small values of *J*, the configurations that dominate the partition function *Z* of Eq. have minimal values of all dual variables,\n \n$$\\begin{aligned}\n \\quad k\\_{x,\\nu}=\\overline{m}\\_{x,\\nu}=0 \\quad \\forall x, \\nu \\quad \\mbox{and} \\quad m\\_{x,1} = 0\n \\quad \\forall x~,\\end{aligned}$$ \nexcept the temporal component of the flux variable, which assumes a constant value\n \n$$\\begin{aligned}\n m\\_{x,2}={r}\\quad \\forall x~,\\end{aligned}$$ \nwhich amounts to *r* static particle world lines on each temporal bond. From Eq. it is clear that the resulting particle density is *n* = *r* (in lattice units). We restrict ourselves to positive *μ* and thus positive *r* for simplicity. These configurations obey the constraints in Eq. and result in the partition functions\n \n$$\\begin{aligned}\n Z\\_{r}(J,\\mu)=\\left[\\left(\\frac{Je^{\\mu}}{2}\\right)^{r}\\,\\frac{\\Gamma(1/2)}{\\Gamma(r+3/2)}\\right]^{N\\_sN\\_t},\\end{aligned}$$ \n(where the factorials have cancelled against Γ-factors from the beta function) or grand potential densities\n \n$$\\begin{aligned}\n \\Omega\\_{r}(J,\\mu)=-{r}\\ln\\big(\\frac{Je^{\\mu}}{2}\\big)+\\ln\\Gamma({r}+3/2) \\;,\\end{aligned}$$ \nup to an irrelevant additive constant. Certain values of *r* yield the smallest grand potential Ω*r* depending on the values of *μ* and *J*. 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Then the *I**i*-shift can be transfered to a uniform shift in the *k**i*, because the Δ-term contains differences of *k*’s only and thus is unchanged.[↩](#fnref3)\n4. Because of this, the equations are only exact at zero temperature. Nevertheless we will use them at finite, but small temperature to compare to lattice data.[↩](#fnref4)\n"},"data_id":{"kind":"string","value":"arxiv_0182785"}}},{"rowIdx":45697,"cells":{"content":{"kind":"string","value":" by at least two orders of magnitude. We are planning a systematic search for other systems presenting these particular features.\nEpisode 2 is identified as a canonical long GRB which originates from the black hole formation process and lasts in arrival time from 50 s to 151 s after the trigger time. The good quality of data allowed us to search for the P-GRB signature in the early emission of the episode 2. From a detailed analysis we find that the first 4 s of episode 2 are in good agreement with the theoretically predicted P-GRB emission, see section 5.2. The observed spectrum integrated over these 4 s is well fitted by a blackbody with an extra power-law component, where this latter component is mainly due to the early emission of the extended-afterglow, see Fig. [fig:pgrb]. From the temperature observed in the P-GRB, *k**T**P**G**R**B* = 29.22 ± 2.21, and the *E**t**o**t**e*+*e*− energy of the second episode, which we assumed equal to the isotropic equivalent energy of this episode, *E**t**o**t**e*+*e*− = 2.49 × 1053 ergs, we obtained the value of the baryon load of the GRB, see also Fig. [fig:no4], *B* = (1.98 ± 0.15) × 10− 3, and a consequent Lorentz Γ factor at the transparency of Γ∘ = 495 ± 40. We have been able to simulate the temporal and the spectral emission of the second episode, as seen by the Fermi-GBM instrument (8 keV – 10 MeV). As we have shown in Fig. [fig:comp], our simulation succeeds in fitting the light curves as well as the spectral energy distribution emitted in the first main spike of the second episode. The residual emission of the last spikes is reasonably fitted, taking into due account the difficulties in integrating the equations of motion, which after the first interactions of the fireshell with the CBM become hardly predictable. The energetic of the simulation is fulfilled and we find that the emission is due to blobs of matter in the CBM with typical dimensions of *r**b**l* = 1016 cm and average density contrast *δ**n*/*n* ≃ 2 particles/cm3 in an overall average density of 1 particle/cm3. We need to find additional cases of such phenomena to augment our statistic and improve its comprehension.\nParticularly relevant are the first two-dimensional hydrodynamical simulations of the progenitor evolution of a $23\nM\\_{\\sun}$ star close to core-collapse, leading to a naked core, as shown in the recent work of Arnett and Meakin. In that work, pronounced asymmetries and strong dynamical interactions between burning shells are seen: the dynamical behavior proceeds to large amplitudes, enlarging deviations from the spherical symmetry in the burning shells. It is of clear interest to find a possible connection between the proto black hole concept, introduced in this work, with the Arnett and Meakin results: to compare the radius, the temperature and the dynamics of the core we have found in the present work with the naked core obtained by Arnett and Meakin from the thermonuclear evolution of the progenitor star. Particularly relevant is the presence, during this phase of collapse, of strong waves, originated in the mixing of the different element’ shells. Such waves should become compressional, as they propagate inward, but they should also dissipate in non-convective regions, causing heating and slow mixing in these regions of the star. Since the wave heating is faster than radiative diffusion (which is very slow), an expansion phase of the boundary layers will occur, while the iron (Fe) core will contract. There is also the interesting possibility that the CBM clouds observed in GRBs be related to the vigorous dynamics in violent activity of matter ejected in the evolution of the original massive star, well before the formation of the naked core (Arnett D., private communication).\nIt is appropriate to emphasize that these results have no relation with the study of precursors in GRBs done in the current literature. Episode 1 and episode 2 are not temporally separated by a quiescent time. The spectral feature of episode 1 and episode 2 are strikingly different and, moreover, the episode 1 is very energetic, which is quite unusual for a typical precursor event. We finally conclude that for the first time we witness the process of formation of the black hole from the phases just preceding the gravitational collapse all the way up to the GRB emission.\nThere is now evidence that the Proto Black Hole formation has been observed also in other GRB sources. After the submission of this article a second example has been found in GRB 101023, then and a paper about this source was submitted on November 4th 2011 and then published on February 1st 2012. There, extremely novel considerations in the structure of the late phase of the emission in X-ray at times larger than 200 s have been presented in favour of a standard signature in these sources. The possible use of this new family of GRBs as distance indicators is being considered.\nWe thank David Arnett for most fruitful discussions, the participants of the Les Houches workshop “From Nuclei to White Dwarfs and Neutron Stars” held in April 2011 (Eds. A. Mezzacappa and R. Ruffini, World Scientific 2011, in press), as well as the members of the AlbaNova University High Energy Astrophysics group. We are thankful to an anonymous referee for her/his important remarks both on the content and the presentation of our work which have improved the presentation of our paper. LI is especially grateful to Marco Muccino for fruitful discussions about the work concerning this manuscript. We are also greateful to the Swift and Fermi teams for their assistance. One of us, AVP, acknowledges the support for the fellowship awarded for the Erasmus Mundus IRAP PhD program. 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A., Svensson, R., & Paczynski, B. 1991,, 366, 343\n---\n1. http://heasarc.gsfc.nasa.gov/lheasoft/[↩](#fnref1)\n2. ftp://legacy.gsfc.nasa.gov/swift/data/obs/[↩](#fnref2)\n3. ftp://legacy.gsfc.nasa.gov/fermi/data/gbm/[↩](#fnref3)\n\nA double component in GRB 090618: a proto-black hole and a genuinely long GRB\n=============================================================================\nIntroduction\n============\nAfter the discovery of the GRBs by the Vela satellites, the first systematic analysis on a large sample of GRBs was possible thanks to the observations of the BATSE instrument on board the Compton Gamma-Ray Observer (CGRO) satellite. The 4BATSE catalog consists of 2704 confirmed GRBs, and it is widely used by the science community as a reference for spectral and timing analysis on GRBs. One of the outcomes of this early analysis of GRBs led to the classification of GRBs as a function of their observed time duration. *T*90 was defined as the time interval over which the 90% of the total BATSE background-subtracted counts are observed. The distribution of the *T*90 duration was bi-modal: the GRBs with *T*90 less than 2s were classified as “short” while the ones with *T*90 larger than 2s were classified as “long”.\nAfter the success of BATSE, a large number of space missions dedicated to the GRB observations were launched. Particularly significant was the discovery of an additional prolonged soft X-ray emission by Beppo-SAX, following the usual hard X-ray emission observed by BATSE. The Beppo-SAX observed emission was named as the “afterglow”, while the BATSE one as the “prompt” radiation. The afterglow allowed to pinpoint more accurately the GRB position in the sky and permitted the identification of their optical counterpart by space and ground based telescopes. The measurement of the cosmological redshift for GRBs became possible and their cosmological nature was firmly established.\nThe Beppo-SAX and related results led to rule out literally hundreds of theoretical models of GRBs,. Among the handful of surviving models, there was the one by based on the mass-energy formula of Black Holes. This model can naturally explain the energetics up to 1054 − 55 erg, as requested by the cosmological nature of GRBs, through the creation of an *e*+*e*−-plasma by vacuum polarization processes in the Kerr-Newman geometry. This model was proposed a few months after the presentation of the discovery of GRBs by Strong at the AAAS meeting in San Francisco.\nIt soon became clear that, as suggested by Goodman and Paczynski, the presence of a Lorentz gamma factor larger than 100 could overcome the problem of opacity of the *e*+*e*−-plasma and justify the *γ*-ray emission of GRBs at cosmological distances. That the dynamics of an *e*+*e*−-plasma with a baryon load with mass *M**B* would naturally lead to Lorentz gamma factor in the range (102- 103) was demonstrated by. The general solution for a baryon load *B* = *M**B**c*2/*E**t**o**t**e*+*e*− between 0 and 10− 2 was obtained in. The interaction between the accelerated baryons with the CBM, indicated by, was advocated to explain the nature of the afterglow.\nThe unprecedented existence of such large Lorentz gamma factors led to formulate the Relativistic Space-Time Transformations paradigm for GRBs. Such a paradigm made it a necessity to have a global, instead of a piecewise, description of a GRB phenomenon. This global description led to the conclusion that the emission by the accelerated baryons interacting with the CBM indeed occurs already in the prompt emission phase in a fully radiative regime. A new interpretation of the burst structure paradigm was then introduced : the existence of a characteristic emission at the transparency of an *e*+*e*−-plasma, the Proper-GRB, followed by an extended-afterglow emission. The relative intensity of these two components is a function of the baryon load. It was proposed that the case of *B* < 10− 5 corresponds to the short GRBs, while the case of *B* > 3 × 10− 4 corresponds to the long GRBs.\nThis different parametrization of the prompt – afterglow versus the one of the P-GRB – extended-afterglow could have originated years of academic discussions. However a clear cut observational evidence came from the Swift satellite, in favour of the second parametrization. The Norris-Bonnell sources, characterized by an initial short spike-like emission in the hard X-rays followed by a softer extended emission, had been indicated in the literature as short bursts. There is a clear evidence that they belong to a new class of “disguised” short GRBs,, where the initial spike is identified as the P-GRB while the prolonged soft emission occurring from the extended-afterglow emission in a CBM typically of the galatic halo. These sources have a baryon load 10− 4 < *B* < 7 × 10− 4: they are just long GRBs exploding in a particularly low density CBM of the order of 10− 3 particles/cm3. This class of sources has given the first evidence of GRBs originating from binary mergers, strongly supported also from direct optical observations.\nIt is interesting that, independent of the development of new missions, the BATSE data continue to attract full scientific interests, even after the end of the mission in the 2000. Important inferences, based on the BATSE data, on the spectra of the early emission of the GRB have been made by and. They have convincingly demonstrated that the spectral feature composed by a blackbody and a power-law plays an important role in selected episodes in the early part of the GRB emission. They have also shown, in some cases, a power-law variation of the thermal component as a function of time, following a broken power-law behavior, see Fig. [fig:no17].\nThe arrival of the Fermi and other satellites allowed further progresses in the understanding of the GRB phenomenon in a much wider energy range. Thanks to the Gamma-Ray Burst Monitor (GBM) and the Large Area Telescope (LAT), additional data are obtained in the 8 keV - 40 MeV and 100 MeV - 300 GeV energy range. It has allowed, among others, this first evidence of a GRB originating from the collapse of a core in the late evolution of a massive star, what we have called the Proto Black Hole.\nIn the specific case of GRB 090618, it has been possible to obtain a complete temporal coverage of the emission in gamma and X-rays, due to the joint observations by Swift, Fermi, AGILE, RT-2/Coronas-PHOTON, Konus-WIND and Suzaku-WAM telescopes. A full coverage in the optical bands, up to 100 days from the burst trigger, has been obtained. This has allowed the determination of the redshift, *z* = 0.54, of the source from spectroscopical identification of absorption lines and a recent claim of a possible supernova emission ∼ 10 days after the GRB trigger. This GRB lasts for ∼ 150 s in hard X-rays, and it is characterized by four prominent pulses. In the soft X-rays there are observations up to 30 days from the burst trigger.\nWe have pointed out in that two different episodes are present in GRB 090618. We have also showed that while the second episode may fit a canonical GRB, the first episode is not expected to be either a part of a GRB or an independent GRB.\nIn the present paper we enter in the merit of the nature of these two episodes. In particular:\n* in Section 2, we describe the observations and data reduction and analysis. We obtain the Fermi GBM (8 keV - 1 MeV and 260 keV - 40 MeV) flux light curves, shown in Fig. [fig:1], following the standard data reduction procedure, and make a detailed spectral analysis of the main emission features, using a Band and a power-law with an exponential high energy cut-off spectral models.\n* in Section 3, after a discussion about the most quoted GRB model, the fireball, we recall the main features of the fireshell scenario, focusing on the reaching of transparency at the end of the initial optically thick phase, with the emission of the Proper-GRB (P-GRB). In Fig. [fig:no4g] we give the theoretical evolution of the Lorentz Γ factor as a function of the radius, for selected values of the baryon load, corresponding to fixed values of the total energy *E**t**o**t**e*+*e*−. The identification of the P-GRB is crucial in determining the main fireshell parameters, which describe the canonical GRB emission. The P-GRB emission is indeed characterized by the temperature, the radius and the Lorentz Γ factor at the transparency, which are related with the *E**t**o**t**e*+*e*− energy and the baryon load, see Fig. [fig:no4]. We then recall the theoretical treatment, the simulation of the light curve and spectrum of the extended-afterglow and, in particular, the determination of the equations of motion, the role of the EQuiTemporal Surfaces (EQTS), as well as the ansatz of the spectral energy distribution in the fireshell comoving frame,.\nThe temporal variability of a GRB light curve has been interpreted in some current models as due to internal shock. In the fireshell model instead such temporal variability is produced by the interaction of the ultra-relativistic baryons colliding with the inhomogeneities of the CircumBurst Medium (CBM). This allows to perform a tomography of the CBM medium around the location of the black hole formation, see Fig. [fig:rad], gaining important information on its structure. These collisions are described by three parameters: the *n**C**B**M* average density, the filling factor R, the clumpiness on scales of 1015 − 16 cm and average density contrast $10^{-1} \\lesssim \\langle\\delta n/n\\rangle \\lesssim 10$. We then refer also to the explanation of the observed hard-to-soft behavior due to the drop of the Lorentz Γ factor and the curvature effect of the EQTS. We then recall the determination of the instantaneous spectra and the simulations of the observed multi-band light curves in the chosen time interval, taking into account all the thousands of convolutions of comoving spectra over each EQTS leading to the observed spectrum. We also emphasize how these simulations have to be performed together and optimized.\n* in Section 4, we perform a spectral analysis of GRB 090618. We have divided the total GRB emission in 6 time intervals, see Table [tab:no1], each one identifying a significant feature in the emission process, see also. We have considered two different spectral models in the data fitting procedure: a Band model and one by a blackbody plus a power-law component, following e.g.. We find that the first 50s of emission are well-fitted by both models, equally the following 9s, from 50 to 59s. The remaining part, from 59 to 151 s, is fitted satisfactorily only by the Band model, see Table [tab:no1].\n* in Section 5, we proceed to the analysis of GRB 090618 in the fireshell scenario. In Section 5.1, we attempt our first interpretation of GRB 090618 assuming it to be a single GRB. We recall that the blackbody is an expected feature in the theory of P-GRB. From the spectral analysis of the first 50 s, we find a spectral distribution consistent with a blackbody plus a power-law component. We have first attempted a fit of the source identifying these first 50s as the P-GRB, see Fig. [fig:big1]. We confirm the conclusion reached in that this interpretation is not sustainable for three different reasons, based on: 1) the energetics of the source, 2) the time duration and 3) the theoretical expected temperature for the P-GRB. We then proceed, in the sub-section 5.2, to an interpretation of GRB 090618 as a multi component system, following the procedure outlined in, in which we outline the possibility of the second episode between 50 and 151s to be an independent GRB.\nWe identify the P-GRB of this second episode, as the first 4s of emission. We find that the spectrum in this initial emission can be fitted by a blackbody plus a power-law component, see Fig. [fig:pgrb]. Since this extra power-law component can be due to the early onset of the extended-afterglow, we take it into account to perform a fireshell simulation which is shown in Fig. [fig:pgrb], with an energy *E**t**o**t**e*+*e*− = 2.49 × 1053 erg and a baryon load *B* = 1.98 ± 0.15 × 10− 3. In Figs. [fig:rad],[fig:firesh],[fig:comp], we report the results of our simulations, summarized in Table [tab:ris]. We notice, in particular, the presence of a strong time lag in this GRB. A detailed analysis, see, about the time lags in the mean energy ranges of 35 keV, 68 keV and 125 keV, reports a quite large lag, ∼ 7 s, in the first 50s of the emission which is unusual for GRBs, while in the following emission, from 51 to 151 s, the observed lags are quite normal, ∼ 1 s.\n* in Section 6, we perform a spectral analysis of the first 50s, where we find a strong spectral variation with time, as reported in Table [tab:no] and in Figs. [fig:no6],[fig:no17], with a chacteristic power-law time variation similar to the ones identified by in a sample of 49 BATSE GRBs.\n* in Section 7, we estimate the variability of the radius emitter, Fig. [fig:no18], and proceed to an estimate of the early expansion velocity. We interpret this data as originating in the expansion process occurring previous to the collapse of the core of a massive star to a black hole, see e.g. : this early 50s of the emission are then defined as the proto-black hole phenomenon.\n* In Section 8, we proceed to the conclusions.\nObservations\n============\nOn 18th of June 2009, the Burst Alert Telescope (BAT) on board the Swift satellite triggered on GRB 090618. After 120 s the X-Ray Telescope (XRT) and the UltraViolet Optical Telescope (UVOT) on board the same satellite, started the observations of the afterglow of GRB 090618. UVOT found a very bright optical counterpart, with a white filter magnitude of 14.27 ± 0.01 not corrected for the extinction, at the coordinates RA(J2000) = 19:35:58.69 = 293.99456, DEC(J2000) = +78:21:24.3 = 78.35676. The BAT light curve shows a multi peak structure, whose total estimated duration is of ∼ 320 s, with the T90 duration in the (15-350) keV range was of 113 s. The first 50 s of the light curve presents a smooth decay trend, followed by a spiky emission, with three prominent peaks at 62, 80 and 112 seconds after the trigger time, respectively, and each have the typical appearance of the FRED pulse (see e.g. ), see Fig. [fig:1]. The time integrated spectrum, (t0 - 4.4, t0 + 213.6) s in the (15-150)keV range, was found to be in agreement with a power-law spectral model with an exponential cutoff, whose photon index was *γ* = 1.42 ± 0.08 and a cut-off energy *E**p**e**a**k* = 134 ± 19 keV. The XRT observations started 125 s after the BAT trigger time and lasted ∼ 25.6 ks and reported an initially bright uncatalogued source, identified as the afterglow of GRB 090618. Its early decay was very steep, ending at 310 s after the trigger time, when it starts a shallower phase, the plateau. Then the light curve breaks to a more steep last phase.\nGRB 090618 was observed also by the Gamma-ray Burst Monitor (GBM) on board the Fermi satellite. From a first analysis, the time-integrated spectrum, (*t*0, *t*0 + 140) s in the (8-1000)keV range, was fitted by a Band spectral model, with a peak energy *E**p**e**a**k* = 155.5 keV, *α* = − 1.26 and *β* = − 2.50, but with strong spectral variations within the considered time interval.\nIt is appropriate to compare and contrast the considerations of the time-integrated spectral analysis, often adopted in the current literature of GRBs, with the information from the time-resolved spectral analysis, as presented e.g. in this article. For a traditional astrophysical source, steady during the observation time, the time-integrated and time-resolved spectral analysis usually coincide. In the case of GRBs, although the duration is only a few seconds, each instantaneous observation corresponds to a very different physical process and the two approaches have an extremely different physical and astrophysical content.\nThe redshift of the source is *z* = 0.54 and it was determined thanks to the identification of the MgII, Mg I and FeII absorption lines, using the KAST spectrograph mounted at the 3-m Shane telescope at the Lick observatory. Given the redshift, and the distance of the source, we computed the emitted isotropic energy in the 8 - 10000 keV energy range, using the Schaefer formula : using the fluence in the (8-1000 keV) as observed by Fermi-GBM, S*o**b**s* = 2.7 × 10− 4, and the ΛCDM cosmological standard model *H*0 = 70 km/s/Mpc, Ω*m* = 0.27, ΩΛ = 0.73, we obtain for the isotropic energy emitted the value of E*i**s**o* = 2.90 × 1053 erg.\nThis GRB was observed also by Konus-WIND, Suzaku-WAM and by the AGILE satellite, which detected emission in the (18-60) keV and in the MCAL instrument, operating at energies greater than 350 keV, but it did not observe high energy photons above 30 MeV. GRB 090618 was the first GRB observed by the Indian payloads RT-2 on board Russian Satellite CORONAS-PHOTON. Two detectors, namely, RT-2/S and RT-2/G consist of NaI(Tl)/CsI(Na) scintillators in phoswich assembly viewed by a photomultiplier tube (PMT). RT-2/S has a viewing angle of 4∘ × 4∘ and covers an energy range of 15 keV to 1 MeV, whereas RT-2/G has an Al filter which sets the lower energy to ∼ 20keV. The Mission was launched from Plesetsk Cosmodrom, Russia on January 30, 2009. During the event RT-2 payload was in the SHADOW mode (away from the Sun) during 08:16:10.207 UT and ended at 08:37:35.465 UT and the GRB 090618 was detected at 77∘ off-axis angle. During this period, the spectra was accumulated in every 100s while the eight channel count rates for each detector are accumulated every second. The entire episode was observed for a duration of more than 200 seconds. A closer examination of the data in the accumulated Channels 1:15-102 keV, 2:95-250 keV and 3:250-1000 keV indicates that the most significant counts is in Channel 2 with a clear evidence of the followings: (a) The emission in the first 50 s is prominent and broader in the lower channels, see Fig. [fig:Chak], (b) After the first 50 s, there is an evidence of a precursor of about 6 seconds duration before the main pulse (c) a break up into two peaks of the main pulse at intermediate energies (35-200 keV) while at higher energies (250-1000 keV) only the first peak of the main pulse survives, see and also Fig. [fig:1] in this manuscript.\nThanks to the complete data coverage of the optical afterglow of GRB 090618, the possible presence of a supernova underlying the emission of the GRB 090618 optical afterglow was reported. The evidence of a supernova emission came from the presence of several bumps in the light curve and by the change in *R**c* - *i* color index over time: in the early phases, the blue color is dominant, typical of the GRB afterglow, but then the color index increases, suggesting a presence of a core-collapse SN. At late times, the contribution from the host galaxy was dominant.\n [fig:Chak]\nData Analysis\n-------------\nWe consider the BAT and XRT data of the Swift satellite, together with the Fermi-GBM and RT2 data of the Coronas-PHOTON satellite. The data reduction was done using the Heasoft v6.10 packages[1](#fn1) for BAT and XRT, and the Fermi-Science tools for GBM.\nWe obtained the BAT light curve and spectra using the standard `headas` procedure. After the data download from the `gsfc` website[2](#fn2), we made a detector quality map and corrected the event data for the known errors of the detector and the hot pixels. We subtracted the background from the data, corrected for the improved position, using the tool `batmaskwtevt` and obtained the 1-s binned light curves and spectra in the main BAT energy band 15 - 150 keV and its subranges, using the tool `batbinevt`. After the systematic corrections to the spectrum, we created the response matrices and obtained the final spectra.\nFor the XRT data, we obtained a total dataset using the standard pipeline, while for a time-resolved analysis we considered the on-line recipe, which is well described in literature, see. The GBM data[3](#fn3), in particular the fourth NaI detector in the (8 - 440 keV) and the b0 BGO detector (260 keV - 40 MeV), were analyzed using the `gtbindef` tool to obtain a GTI file for the energy distribution and the `gtbin` for the light curves and final spectra. In order to obtain an energy flux lightcurve, we made a time resolved spectral analysis dividing the count lightcurve in six time intervals, each of them corresponding to a particular pulse, as described in the work of. All the time resolved spectra were fitted using the XSPEC data analysis software version 12.6.0q, included in the Heasoft data package, and considering for each spectrum a classical Band spectral model and a power-law model with an exponential energy cut-off, folded through the detector response matrix. After the subtraction of the background, we fit the spectrum by minimizing the *χ*2 between the spectral models described above and the observed data, obtaining the best-fit spectral parameters and the respective model normalization. In Table [tab:no1] we give the results of our spectral analysis. The time reported in the first column corresponds to the time after the GBM trigger time t*t**r**i**g* = 267006508 s, where the *β* parameter was not constrained, we used its averaged value, as delineated in *β* = -2.3 ± 0.10. We have considered the chi-square statistic for testing our data fitting procedure. The reduced chi-square *χ̃*2 = *χ*2/*N*, where *N* is the number of degrees of freedom (dof) which is *N* = 82 for the NaI dataset and *N* = 121 for the BGO one.\nFor the last pulse of the second episode, the Band model is not very precise (*χ̃*2 = 2.24), but a slightly better approximation is given by the power-law with an exponential cut-off, whose fit results are shown for the same intervals in the last two columns. From these values, we build the flux light curves for both the detectors, which are shown in fig. [fig:1].\n| |\n| --- |\n| |\n[fig:1]\n[tab:no1]\n| Time Interval | *α* | *β* | *E*0*B**A**N**D* (keV) | *χ̃**B**A**N**D*2 | *γ* | *E*0*c**u**t* (keV) | *χ̃**c**u**t*2 |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 0 - 50 | -0.77− 0.28+ 0.38 | -2.33− 0.28+ 0.33 | 128.12− 56.2+ 109.4 | 1.11 | 0.91− 0.21+ 0.18 | 180.9− 54.2+ 93.1 | 1.13 |\n| 50 - 57 | -0.93− 0.37+ 0.48 | -2.30 ± 0.10 | 104.98− 51.7+ 142.3 | 1.22 | 1.11− 0.30+ 0.25 | 168.3− 70.2+ 158.6 | 1.22 |\n| 57 - 68 | -0.93− 0.08+ 0.09 | -2.43− 0.67+ 0.21 | 264.0− 54.4+ 75.8 | 1.85 | 1.01− 0.06+ 0.06 | 340.5− 45.4+ 56.0 | 1.93 |\n| 68 - 76 | -1.05− 0.07+ 0.08 | -2.49− 0.49+ 0.21 | 243.9− 53.0+ 57.1 | 1.88 | 1.12− 0.04+ 0.04 | 311.0− 32.9+ 38.6 | 1.90 |\n| 76 - 103 | -1.06− 0.08+ 0.08 | -2.65− 0.34+ 0.19 | 125.7− 19.26+ 23.27 | 1.23 | 1.15− 0.06+ 0.06 | 157.7− 18.6+ 22.2 | 1.39 |\n| 103 - 150 | -1.50− 0.18+ 0.20 | -2.30 ± 0.10 | 101.1− 30.5+ 58.3 | 1.07 | 1.50− 0.20+ 0.18 | 102.8− 30.4+ 56.8 | 1.06 |\nWe turn now to the XRT which started to observe GRB 090618 ∼ 120 s after the BAT trigger. Its early data show a continued activity of the prompt emission, fading away ∼ 200 s after the BAT trigger time. Then the light curve is well approximated with a power-law decay. In view of the lack of soft X-ray data before the onset of the XRT, we cannot exclude a previous pulse in the X-ray light curve emission of GRB 090618. The following shallow and late decay phases, well-known in literature, will not be analyzed in this paper since we focus in the first 200 s of the GRB emission.\nA brief review of the fireshell and alternative models\n======================================================\nThe GRB prompt emission in the fireball scenario\n------------------------------------------------\nA variety of models have been developed to theoretically explain the observational properties of GRBs. One of the most quoted is the fireball model (see for a review ). The model was first proposed by, and, who have shown that the sudden release of a large quantity of energy in a compact region can lead to an optically thick photon-lepton plasma and to the production of *e*+*e*− pairs. The total annihilation of the *e*+*e*− plasma was assumed, leading to a vast release of energy pushing on the CBM: the “fireball”.\nAn alternative approach, originating in the gravitational collapse to a black hole, is the fireshell model (see for a review and ). There the GRBs originate from an optically thick electron–positron plasma in thermal equilibrium, having a total energy of *E**t**o**t**e*±. Such plasma is initially confined between the radius of a black hole *r**h* and the dyadosphere radius\n \n$$\\label{eq:rh}\nr\\_{ds}= r\\_h \\left[2 \\alpha \\frac{E\\_{tot}^{e^+e^-}}{m\\_e c^2}\\left(\\frac{\\hbar /m\\_e c}{r\\_h}\\right)^3\\right]^{1/4},$$ \nwhere, *α* is the usual fine structure constant, ℏ and *c* the Planck constant and the speed of light, and *m**e* the mass of the electron. The lower limit of *E**t**o**t**e*± coincides with *E**i**s**o*. The condition of thermal equilibrium assumed in this model as shown by, differentiates this approach from the alternative ones, see Section 3.2.\nIn the fireball model, the prompt emission, including the sharp luminosity variations are due to the prolonged and variable activity of the “inner engine”. The conversion of the fireball energy to radiation originates by shocks, either internal (when faster moving matter takes over a slower moving shell, see ) or external (when the moving matter is slowed down by the external medium surrounding the burst, see ). Much attention has been given to the Synchrotron emission from relativistic electrons, possibly accompanied by SSC emission to explain the observed GRB spectrum. These processes were found to be consistent with the observational data of many GRBs. However, several limitations have been evidenced in relation with the low energy spectral slopes of time-integrated spectra and time-resolved spectra. Additional limitations on SSC have also been pointed out by and.\nThe latest phases of the afterglow are described in the fireball model by assuming an equation of motion given by the Blandford-McKee self-similar power-law solution. The maximum Lorentz factor of the fireball is estimated from the temporal occurrence of the peak of the optical emission, which is identified with the peak of the forward external shock emission in the thin shell approximation. There have been developed partly alternative and/or complementary scenarios to the fireball model, e.g. the ones based on: quasi-thermal Comptonization, Compton drag emission, Synchrotron emission from a decaying magnetic field, jitter radiation, Compton scattering of synchrotron self absorbed photons, photospheric emission. In particular, pointed out that the photospheric emission overcomes some of the difficulties of pure non-thermal emission models.\nThe fireshell scenario\n----------------------\nIn the fireshell model, the rate equation for the *e*+*e*− pairs and its dynamics have been given by (the pair-electromagnetic pulse or PEM pulse for short). This plasma engulfs the baryonic material left over in the process of gravitational collapse having mass *M**B*, still keeping thermal equilibrium between electrons, positrons and baryons. The baryon load is measured by the dimensionless parameter *B* = *M**B**c*2/*E**t**o**t**e*+*e*−. It was shown that no relativistic expansion of the plasma can be found for *B* > 10− 2. The fireshell is still optically thick and self-accelerates to ultrarelativistic velocities. Then the fireshell becomes transparent and the (P-GRB) is emitted. The final Lorentz gamma factor at transparency can vary in a vast range between 102 and 103 as a function of *E**t**o**t**e*+*e*− and *B*, see Fig. [fig:no4g]. For the final determination it is necessary to integrate explicitly the rate equation of the *e*+*e*− annihilation process and evaluate, for a given black hole mass and a given *e*+*e*− plasma radius, the reaching of the transparency condition, see Fig. [fig:no4].\nThe fireshell scenario does not require any prolonged activity of the inner engine. After transparency, the remaining accelerated baryonic matter still expands ballistically and starts to slow down by the collisions with the CBM, having average density *n**c**b**m*. In the standard fireball scenario, the spiky light curve is assumed to be caused by internal shocks. In the fireshell model the entire extended-afterglow emission is assumed to originate from an expanding thin shell enforcing energy and momentum conservation in the collision with the CBM. The condition of a fully radiative regime is assumed. This, in turn, allows to estimate the characteristic inhomogeneities of the CBM, as well as its average value.\nIt is appropriate to recall a further difference between our treatment and the ones in the current literature. The complete analytic solution of the equations of motion of the baryonic shell has been developed, while in the current literature usually the Blandford - McKee self-similar solution has been uncritically adopted. The analogies and differences between the two approaches have been explicitly pointed out in.\nFrom this general approach, a canonical GRB bolometric light curve composed of two different parts is defined: the P-GRB and the extended-afterglow. The relative energetics of these two components, the observed temporal separation between the corresponding peaks, is a function of the above three parameters *E**t**o**t**e*+*e*−, *B*, and the average value of the *n**c**b**m*; the first two parameters are inherent to the accelerator characterizing the GRB, i.e., the optically thick phase, while the third one is inherent to the GRB surrounding environment which gives rise to the extended-afterglow. If one goes to the observational properties of this model of a relativistic expanding shell, a crucial concept has been the introduction of the EQTS. In this topic, also, our model differs from the ones in the literature for deriving the analytic expression of the EQTS from the analytic solutions of the equations of motion.\nIn this paper, we assume *E**t**o**t**e*+*e*− = *E**i**s**o*. This assumption is based on the very accurate information we have on the luminosity and the spectral properties of the source. In other GRBs, we have assumed *E**t**o**t**e*+*e*− > *E**i**s**o* to take into account the observational limitations, due to detector thresholds, distance effects and lack of data.\nThe emission of the P-GRB\n-------------------------\nThe lower limit of *E**t**o**t**e*+*e*− is given by the observed isotropic energy emitted in the GRB, *E**i**s**o*. The identification of the energy of the afterglow and of the P-GRB determines the baryon load *B* and, from these, it is possible to determine, see Fig. [fig:no4]: the value of the Lorentz Γ factor at transparency, the observed temperature as well as the temperature in the comoving frame and the laboratory radius at transparency. We can determine indeed from the spectral analysis of the P-GRB candidate, the temperature *k**T**o**b**s* and the energy emitted in the transparency *E**P**G**R**B*. The relation between these parameters can not be expressed by an analytical formulation: they can be only obtained by a numerical integration of the entire fireshell equations of motion. In practice we need to perform a trial and error procedure to find the set of values which fit the observations.\n [fig:no4g]\n(a) \n \n(b) \n \n(c) \n \n(d) \n \n [fig:no4]\nAs we are going to see in the case of GRB 090618, the direct measure of the temperature of the thermal component at the transparency offers a very important new information in the determination of the GRB parameters. In the emission of the P-GRB two different phases are present: one corresponding to the emission of the photons when the transparency is reached, and the second is the early interaction of the ultra-relativistic protons and electrons with the CBM. A spectral energy distribution with a thermal component and a non-thermal one should be expected to occur.\nThe extended-afterglow\n----------------------\nThe majority of works in the current literature has addressed the analysis of the afterglow emission as due to various combinations of Synchrotron and Inverse Compton processes, see e.g.. It appears, however, that this description is not fully satisfactory.\nWe have adopted in the fireshell model a pragmatic approach by making the full use of the knowledge of the equations of motion, of the EQTS formulations as well as of the correct relativistic transformations between the comoving frame of the fireshell and the observer frame. These equations, that relate the four time variables, are necessary for the interpretation of the GRB data. They are: a) the comoving time, b) the laboratory time, c) the arrival time, and d) the arrival time at the detector corrected by the cosmological effects. This is the content of the Relative Space-Time Transformations paradigm, essential for the interpretation of GRBs data. Such a paradigm made it a necessity to have a global, instead of a piecewise, description of a GRB phenomenon. This global description led to a new interpretation of the burst structure paradigm. As recalled in the Introduction, a new conclusion, arising from the burst structure paradigm, has been that the emission by the accelerated baryons interacting with the CBM is indeed occurring already in the prompt emission phase, just after the P-GRB emission. This is the extended-afterglow emission, which presents in its “light curve” a rising part, a peak, and a decaying tail. Following this paradigm, the prompt emission phase is therefore composed by both the P-GRB emission and the peak of the extended-afterglow.\nTo evaluate the extended-afterglow spectral properties, we have adopted an ansatz on the spectral properties of the emission in the collisions between the baryons and the CBM in the comoving frame. We have then evaluate all the observational properties in the observer frame by integrating on the EQTS. The initial ansatz of thermal spectrum, has been recently modified to\n \n$$\\label{eq:no2}\n\\frac{d N\\_{\\gamma}}{d V d \\epsilon} = \\left(\\frac{8 \\pi}{h^3 c^3}\\right)\\left(\\frac{\\epsilon}{k\\_B T}\\right)^{\\alpha}\\frac{\\epsilon^2}{exp\\left(\\frac{\\epsilon}{k\\_B T}\\right)-1},$$ \nwhere *α* is a phenomenological parameter defined in the comoving frame of the fireshell, determined by the optimization of the simulation of the observed data. It is well known that in the ultrarelativistic collision of protons and electrons with the CBM, collective processes of ultrarelativistic plasma physics are expected, not yet fully explored and understood (e.g. Weibel instability, see ). Promising results along this line have been already obtained by and, and may lead to the understanding of the physycal origin of the *α* parameter in Eq. [eq:no2].\nIn order to take into due account the filamentary, clumpy and porous structure of the CBM, we have introduced the additional parameter R, which describes the fireshell surface filling factor. It is defined as the ratio between the effective emitting area of the fireshell *A**e**f**f* and its total visible area *A**v**i**s*.\nOne of the main features of the GRB afterglow has been the observation of hard to soft spectral variation, which is generally absent in the first spike-like emission, which we have identified as the P-GRB,. An explanation of the hard-to-soft spectral variation has been advanced on the ground of two different contributions: the curvature effect and the intrinsic spectral evolution. In particular, in the work of the authors use the model developed in for the spectral lag analysis, taking into account an intrinsic Band model for the GRBs and a Gaussian profile for the GRB pulses, in order to take into account the angular effects, and they find that both causes provide a very good explanation for the observed time lags. Within the fireshell model we can indeed explain a hard-to-soft spectral variation very naturally, in the extended-afterglow emission. Since the Lorentz Γ factor decreases with time, the observed effective temperature of the fireshell will drop as the emission goes on, so the peak of the emission will occur at lower energies. This effect is amplified by the presence of the curvature effect, which has origin in the EQTS concept. Both these observed features are considered as the responsible for the time lag observed in GRBs.\nThe simulation of a GRB light curve and spectra of the extended-afterglow\n-------------------------------------------------------------------------\nThe simulation of a GRB light curve and the respective spectrum requires also the determination of the filling factor R and of the CBM density *n**C**B**M*. These extra parameters are extrinsic and they are just functions of the radial coordinate from the source. The parameter R, in particular, determines the effective temperature in the comoving frame and the corresponding peak energy of the spectrum, while *n**c**b**m* determines the temporal behavior of the light curve. It is found that the CBM is typically formed of “clumps” of width ∼ 1015 − 16 cm and average density contrast $10^{-1} \\lesssim \\langle \\delta n/n \\rangle \\lesssim 10$ centered on the value of 4 *p**a**r**t**i**c**l**e**s*/*c**m*3, see Fig. [fig:rad], and clumps of masses *M**c**l**u**m**p* ≈ 1022 − 24 g. Particularly important is the determination of the average value of *n**c**b**m*. Values of the order of 0.1-10 particles/cm3 have been found for GRBs exploding inside star forming region galaxies, while values of the order of 10− 3 particles/cm3 have been found for GRBs exploding in galactic halos. The presence of such a clumpy medium, already predicted in pioneering works of Fermi in the theoretical study of interstellar matter in our galaxy, is by now well-established both from the GRB observations and by additional astrophysical observations, see e.g. the circum-burst medium observed in novae, or by theoretical considerations on supergiant, massive stars, clumpy wind. Interesting are the considerations by Arnett and Meakin, who have shown how realistic 2D simulations of the late evolution of a core collapse show processes of violent emission of clouds: there the 2D simulations differ from the one in 1D, which show a much more regular and wind behavior around the collapsing core. Consequently, attention should be given also to instabilities prior to the latest phases of the evolution of the core, possibly giving origin to the cloud pattern observed in the CBM of GRB phenomenon (D.Arnett private communication).\nThe determination of the R and *n**C**B**M* parameters depends essentially on the reproduction of the shape of the extended-afterglow and of the respective spectral emission, in a fixed energy range. Clearly, the simulation of a source within the fireshell model is much more complex than simply fitting the *N*(*E*) spectrum with phenomenological analytic formulas for a finite temporal range of the data. It is a consistent picture, which has to find the best value for the parameters of the source, the P-GRB, its spectrum, its temporal structure, as well as its energetics. For each spike in the light curve are computed the parameters of the corresponding CBM clumps, taking into account all the thousands of convolutions of comoving spectra over each EQTS leading to the observed spectrum. It is clear that, since the EQTS encompass emission processes occurring at different comoving times weighted by their Lorentz and Doppler factors, the “fitting” of a single spike is not only a function of the properties of the specific CBM clump but of the entire previous history of the source. Any mistake at any step of the simulation process affects the entire evolution that follows and, conversely, at any step a fit must be made consistently with all the previous history: due to the non-linearity of the system and to the EQTS, any change in the simulation produces observable effects up to a much later time. This brings to an extremely complex procedure by trial and error in the data simulation, in which the variation of the parameters defining the source are further and further narrowed down, reaching very quickly the uniqueness. Of course, we cannot expect the latest parts of the simulation to be very accurate, since some of the basic hypothesis on the equations of motion, and possible fragmentation of the shell, can affect the procedure.\nIn particular, the theoretical photon number spectrum to be compared with the observational data is obtained by an averaging procedure of instantaneous spectra. In turn, each instantaneous spectrum is linked to the simulation of the observed multiband light curves in the chosen time interval. Therefore, both the simulation of the spectrum and of the observed multiband light curves have to be performed together and simultaneously optimized.\nSpectral analysis of GRB 090618\n===============================\nWe proceed now to the detailed spectral analysis of GRB 090618. We divide the emission in six time intervals, shown in Table [tab:no1], each one identifying a significant feature in the emission process. We then fit for each time interval the spectra by a Band model as well as by a blackbody with an extra power-law component, following. In particular, we are interested in the estimation of the temperature *k**T* and the observed energy flux *ϕ**o**b**s* of the blackbody component. The specific intensity of emission of a thermal spectrum at energy *E* in energy range *d**E* into solid angle ΔΩ is\n \n$$\\label{eq:no2a}\nI(E)dE = \\frac{2}{h^3 c^2} \\frac{E^3}{\\exp(E/kT) - 1} \\Delta \\Omega dE.$$ \nThe source of radius *R* is seen within a solid angle ΔΩ = *π**R*2/*D*2, and its full luminosity is *L* = 4*π**R*2*σ**T*4. What we are fitting however is the background-subtracted photon spectra *A*(*E*), which is obtained by dividing the specific intensity *I*(*E*) by the energy *E*:\n \n$$\\begin{aligned}\nA(E)dE \\equiv \\frac{I(E)}{E}dE &=& \\frac{k^4 L}{2\\sigma (kT)^4 D^2 h^3 c^2}\\frac{E^2 d E}{\\exp(E/kT)-1} \\nonumber\\\\\n&=& \\frac{15 \\phi\\_{obs}}{\\pi^4 (kT)^4}\\frac{E^2 d E}{\\exp(E/kT)-1},\n\\label{eq:no2b}\\end{aligned}$$ \nwhere *h*, *k* and *σ* are respectively the Planck, the Boltzmann and the Stefan-Boltzmann constants, *c* is the speed of light and *ϕ**o**b**s* = *L*/(4*π**D*2) is the observed energy flux of the blackbody emitter. The great advantage of Eq. ([eq:no2b]) is that it is written in terms of the observables *ϕ**o**b**s* and *T*, so from a spectral fitting procedure we can obtain the values of these quantities for each time interval considered. In order to determine these parameters, we must perform an integration of the actual photon spectrum *A*(*E*) over the instrumental response *R*(*i*, *E*) of the detector which observe the source, where *i* denotes the different instrument energy channels. The result is a predicted count spectrum\n \n*C**p*(*i*) = ∫*E**m**i**n*(*i*)*E**m**a**x*(*i*)*A*(*E*)*R*(*i*, *E*)*d**E*, \nwhere *E**m**i**n*(*i*) and *E**m**a**x*(*i*) are the boundaries of the *i*-th energy channel of the instrument. Eq. ([eqint]) must be compared with the observed data by a fit statistic.\nThe main parameters obtained from the fitting procedure are shown in Table [tab:no2b]. We divide the entire GRB in two main episodes, as advanced in : one lasting the first 50 s and the other from 50 to 151 s after the GRB trigger time, see Fig. [fig:cospar]. It is easy to see that the first 50 s of emission, corresponding to the first episode, are well fitted by a Band model as well as a black-body with an extra power-law model, Fig. [fig:big1]. The same happens for the first 9 s of the second episode (from 50 to 59 s after the trigger time), Fig. [fig:big2]. For the subsequent three intervals corresponding to the main peaks in the light curve, the black-body plus a power-law model does not provide a satisfactory fit. Only the Band model fits the spectrum with good accuracy, with the exception of the first main spike (compare the values of *χ*2 in the table). We find also that the last peak can be fitted by a simple power-law model with a photon index *γ* = 2.20 ± 0.03, better than by a Band model.\nThe result of this analysis points to a different emission mechanism in the first 50 s of GRB 090618 and in the following 9 s. A sequence of very large pulses follow, which spectral energy distribution is not attributable either to a blackbody or a blackbody and an extra power-law component. The evidence for the transition is well represented by the test of the data fitting, whose indicator is given by the changing of the $\\tilde{\\chi^2}$ (*N**d**o**f* = 169) for the blackbody plus a power-law model for the different time intervals, see table [tab:no2b]. Although the Band spectral model is an empirical model without a clear physical origin, we do check its validity in all of the time-detailed spectra with the sole exception of the first main pulse of the second episode. The *χ*2 corresponding to the Band model for such a main pulse, although better than the one corresponding to the blackbody and power-law case, is unsatisfactory. We are now going to a direct application of the fireshell model in order to make more stringent the above conclusions and reach a better understanding of the source.\n[tab:no2b]\n| | Time Interval (s) | *α* | *β* | *E*0(*k**e**V*) | *χ̃**B**A**N**D*2 | *k**T*(*k**e**V*) | *γ* | *χ̃**B**B* + *p**o*2 |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| A | 0 - 50 | -0.74 ± 0.10 | -2.32 ± 0.16 | 118.99 ± 21.71 | 1.12 | 32.07 ± 1.85 | 1.75 ± 0.04 | 1.21 |\n| B | 50 - 59 | -1.07 ± 0.06 | -3.18 ± 0.97 | 195.01 ± 30.94 | 1.23 | 31.22 ± 1.49 | 1.78 ± 0.03 | 1.52 |\n| C | 59 - 69 | -0.99 ± 0.02 | -2.60 ± 0.09 | 321.74 ± 14.60 | 2.09 | 47.29 ± 0.68 | 1.67 ± 0.08 | 7.05 |\n| D | 69 - 78 | -1.04 ± 0.03 | -2.42 ± 0.06 | 161.53 ± 11.64 | 1.55 | 29.29 ± 0.57 | 1.78 ± 0.01 | 3.05 |\n| E | 78 - 105 | -1.06 ± 0.03 | -2.62 ± 0.09 | 124.51 ± 7.93 | 1.20 | 24.42 ± 0.43 | 1.86 ± 0.01 | 2.28 |\n| F | 105 - 151 | -2.63 ± -1 | -2.06 ± 0.02 | unconstrained | 1.74 | 16.24 ± 0.84 | 2.23 ± 0.05 | 1.15 |\n[fig:cospar]\nc|c \n[fig:big1]\nc|c \n[fig:big2]\nAnalysis of GRB 090618 in the fireshell scenario: from a single GRB to a multi-component GRB\n============================================================================================\nAttempt for a single GRB scenario: the role of the first episode\n----------------------------------------------------------------\nWe first approach the analysis of GRB 090618 by assuming that we are in presence of a single GRB and attempt to identify its components in a canonical GRB scenario, based on the fireshell model. We first attempt the identification of the P-GRB emission. We have already seen that the integrated first 50 s can be well-fitted with a black-body at a temperature *k**T* = 32.07 ± 1.85 keV and an extra power-law component with the photon index *γ* = -1.75 ± 0.04, see panel A in Fig. [fig:big2] and Table [tab:no2b]. Being the presence of a blackbody component the distinctive feature of the P-GRB, we have first attempted an interpretation of GRB 090618 as a single GRB with the first 50 s as the P-GRB. We have first proceeded to evaluate if the energetics of the emission in the first 50 s can be interpreted as due to a P-GRB. The energy emitted by the sole blackbody is *E**B**B* = 8.35− 0.36+ 0.27 × 1051 ergs. Recalling that the isotropic energy of the entire GRB 090618 is *E**i**s**o* = (2.90 ± 0.02) × 1053 ergs, we have that the blackbody component would be ∼ 2.9 % of the total energy emitted in the burst. This would imply, see lower panel in fig. [fig:no4], a baryon load *B* ∼ 10− 3 with a corresponding Lorentz Γ factor of ∼ 800 and a temperature of ∼ 52 keV. This value is in disagreement with the observed temperature *k**T**o**b**s* = 32.07 keV.\nOne may attempt to reconcile the value of the theoretically predicted GRB temperature with the observed one by increasing *E**t**o**t**e*+*e*−. This would lead to an *E**t**o**t**e*+*e*− = 4 × 1054 ergs and a corresponding baryon load of *B* ≈ 10− 4. This would imply three major discrepancies: a) there would be an unjustified complementary unobserved energy; b) in view of the value of the baryon load, and the corresponding Lorentz Γ factor, the duration of the extended-afterglow emission would be more than an order of magnitude smaller than the observed 100 s ; c) the duration of this first 50 s is much longer than the one typically expected for all P-GRBs identified in other GRBs, which is at maximum of the order of ∼ 10 s. We have therefore considered hopeless this approach and proceeded to a different one looking for multiple components.\nThe multi-component scenario: the second episode as an independent GRB\n----------------------------------------------------------------------\n### The identification of the P-GRB of the second episode\nWe now proceed to the analysis of the data between 50 and 150 s after the trigger time, as a canonical GRB in the fireshell scenario, namely the second episode, see Fig. [fig:cospar],. We proceed to identify the P-GRB within the emission between 50 and 59 s, since we find a blackbody signature in this early second-episode emission. Considerations based on the time variability of the thermal component bring us to consider the first 4 s of such time interval as due to the P-GRB emission. The corresponding spectrum (8-440 keV) is well fitted (*χ̃*2 = 1.15) with a blackbody of a temperature *k**T* = 29.22 ± 2.21 keV (norm = 3.51 ± 0.49), and an extra power-law component with photon index *γ* = 1.85 ± 0.06, (norm = 46.25 ± 10.21), see Fig. [fig:pgrb]. The fit with the Band model is also acceptable (*χ̃*2 = 1.25). The fit gives a low energy power-law index *α* = − 1.22 ± 0.08, a high energy index *β* = − 2.32 ± 0.21 and a break energy *E*0 = 193.2 ± 50.8, see Fig. [fig:pgrb]. In view of the theoretical understanding of the thermal component in the P-GRB, see Section 3.2, we shall focus in the following on the blackbody + power-law spectral model.\nThe isotropic energy of the second episode is *E**i**s**o* = (2.49 ± 0.02) × 1053 ergs. The simulation within the fireshell scenario is done assuming *E**t**o**t**e*+*e*− ≡ *E**i**s**o*. From the upper panel in Fig. [fig:no4] and the observed temperature, we can then derive the corresponding value of the baryon load. The observed temperature of the blackbody component is *k**T* = 29.22 ± 2.21, so that we can determine a value of the baryon load of *B* = 1.98 ± 0.15 × 10− 3, and deduce the energy of the P-GRB as a fraction of the total *E**t**o**t**e*+*e*−. We so obtain a value of the P-GRB energy of 4.33− 0.28+ 0.25 × 1051 erg.\nNow, from the second panel in Fig. [fig:no4] we can derive the radius of the transparency condition, to occur at *r**t**r* = 1.46 × 1014 cm. From the third panel we derive the bulk Lorentz factor of Γ*t**h* = 495. We compare this value with the energy measured in the sole blackbody component of *E**B**B* = 9.24− 0.58+ 0.50 × 1050 erg, and with the energy in the blackbody plus the power-law component of *E**B**B* + *p**o* = 5.43− 0.11+ 0.07 × 1051 erg, and verify that the theoretical value is in between these observed energies. We have found this result quite satisfactory: it represents the first attempt to relate the GRB properties to the details of the black hole responsible for the overall GRB energetics. The above theoretical estimates have been based on a non rotating black hole of 10 M$\\_{\\sun}$, a total energy of *E**t**o**t**e*+*e*− = 2.49 × 1053 erg and a mean temperature of the initial plasma of *e*+*e*− of 2.4 MeV, derived from the expression of the dyadosphere radius, Eq. [eq:rh]. Any refinement of the direct comparison between theory and observations will have to address a variety of fundamental issues such as, for example: 1) the possible effect of rotation of the black hole, leading to a more complex dyadotorus structure; 2) a more detailed analysis of the transparency condition of the *e*+*e*− plasma, simply derived from the condition *τ* = ∫*R**d**r*(*n**e*± + *n**e*−*b*)*σ**T* = 0.67 ; 3) an analysis of the general relativistic, electrodynamical, strong interactions descriptions of the gravitational collapse core leading to a black hole formation,.\nc|c \n[fig:pgrb]\n [fig:pgrb2]\n### The analysis of the extended-afterglow of the second episode\nThe extended-afterglow starts at the above given radius of the transparency, with an initial value of the Lorentz Γ factor of Γ0 = 495. In order to simulate the extended-afterglow emission, we need to determine the radial distribution of the CBM around the burst site, which we assume for simplicity to be spherically symmetric, we infer characteristic size of Δ*R* = 1015 − 16 cm. We have already recalled how the simulate of the spectra and of the observed multi band light curves have to be performed together and jointly optimized, leading to the determination of the fundamental parameters characterizing the CBM medium. This radial distribution is shown in Fig. [fig:rad], and is characterized by a mean value of < *n* > = 0.6 part/cm3 and an average density contrast with a < *δ**n*/*n* > ≈ 2, see Fig. [fig:rad] and Table [tab:dens]. The data up to 8.5 × 1016 cm are simulated with a value for the filling factor R = 3 × 10− 9, while the data from this value on with R = 9 × 10− 9. From the radial distribution of the CBM density, and considering the 1/Γ effect on the fireshell visible area, we found that the CBM clumps originating the spikes in the extended-afterglow emission have masses of the order of 1022 − 24 g. The value of the *α* parameter has been found to be -1.8 along the total duration of the GRB.\nIn Figs. [fig:firesh] we show the simulated light curve (8-1000 keV) of GRB and the corresponding spectrum, using the spectral model described in,.\nWe focus our attention, in particular, on the structure of the first spikes. The comparison between the spectra of the first main spike (t0+59, t0+66 s) of the extended-afterglow of GRB 090618, obtained with three different assumptions is shown in Fig. [fig:comp]: in the upper panel we show the fireshell simulation of the integrated spectrum (t0+59, t0+66 s) of the first main spike, in the middle panel we show the best fit with a blackbody and a power-law component model and in the lower panel the best fit using a simple power-law spectral model.\nWe can see that the fit with the last two models is not satisfactory: the corresponding *χ̃*2 is 7 for the blackbody + power-law and ∼ 15 for the simple power-law. We cannot give the *χ̃*2 of the fireshell simulation, since it is not represented by an explicit analytic fitting function, but it originates by a sequence of complex high non-linear procedure, summarized in Sec. [sec:3]. It is clear by a direct scrutiny that it correctly reproduces the low energy emission, thanks in particular to the role of the *α* parameter, which was described previously. At higher energies, the theoretically predicted spectrum is affected by the cut-off induced by the thermal spectrum. The temporal variability of the first two spikes are well simulated.\nWe are not able to accurately reproduce the last spikes of the light curve, since the equations of motion of the accelerated baryons become very complicated after the first interactions of the fireshell with the CBM. This happens for different reasons. First, a possible fragmentation of the fireshell can occur. Moreover, at larger distances from the progenitor the fireshell visible area becomes larger than the transverse dimension of a typical blob of matter, consequently a modification of the code for a three-dimensional description of the interstellar medium will be needed. This is unlike the early phases in the prompt emission, which is the main topic we address at the moment, where a spherically simmetric approximation applies. The fireshell visible area is smaller than the typical size of the CBM clouds in the early phases of the prompt radiation,.\nThe second episode, lasting from 50 to 151 s, agrees with a canonical GRB in the fireshell scenario. Particularly relevant is the problematic of the P-GRB. It interfaces with the fundamental physics issues, related to the physics of the gravitational collapse and the black hole formation. There is an interface between the reaching of transparency of the P-GRB and the early part of the extended-afterglow. This connection has already been introduced in literature. We have studied this interface in the fireshell by analyzing the thermal emission at the transparency with the early interaction of the baryons with the CBM matter, see Fig. [fig:pgrb2].\nWe turn now to reach a better understanding of the meaning of the first episode, between 0 and 50 s of the GRB emission. To this end we examine the two episodes in respect to: 1) the Amati relation, 2) the hardness variation and 3) the observed time lag.\n [fig:rad]\n[tab:ris]\n| Parameter | Value |\n| --- | --- |\n| *E**t**o**t**e*+*e*− | 2.49 ± 0.02 × 1053 ergs |\n| *B* | 1.98 ± 0.15 × 10− 3 |\n| Γ0 | 495 ± 40 |\n| *k**T**t**h* | 29.22 ± 2.21 keV |\n| *E**P* − *G**R**B*, *t**h* | 4.33 ± 0.28 × 1051 ergs |\n| < *n* > | 0.6 *p**a**r**t*/*c**m*3 |\n| < *δ**n*/*n* > | 2 *p**a**r**t*/*c**m*3 |\n[tab:dens]\n| Cloud | Distance (cm) | r (cm) | *ρ* (#/cm3) | M (g) |\n| --- | --- | --- | --- | --- |\n| First | 4.0 × 1016 | 1 × 1016 | 1 | 2.5 × 1024 |\n| Second | 7.4 × 1016 | 5 × 1015 | 1 | 3.1 × 1023 |\n| Third | 1.1 × 1017 | 2 × 1015 | 4 | 2.0 × 1022 |\n| |\n| --- |\n| |\n| |\n| |\n[fig:firesh]\n| |\n| --- |\n| |\n| |\n| |\n| |\n[fig:comp]\nThe Amati relation, the HR and the time lag of the two episodes\n===============================================================\nThe first episode as an independent GRB?\n----------------------------------------\nWe first check if the two episodes fulfill separately the Amati relation,. By using the Band spectrum we verify that the first episode presents an intrinsic peak energy value of *E**p*, 1*s**t* = 223.01 ± 24.15 keV, while the second episode presents an *E**p*, 2*n**d* = 224.57 ± 17.4 keV. The isotropic energies emitted in each single episode are *E**i**s**o*, 1*s**t* = 4.09 ± 0.07 × 1052 ergs and *E**i**s**o*, 2*n**d* = 2.49 ± 0.02 × 1053 ergs, so we have that both episodes satisfy the Amati relation, see fig. [fig:amati]. The fulfillment of the Amati relation of episode 2 was expected, being the second episode a canonical GRB. What we find surprising is the fulfillment of the Amati relation of the first episode.\n [fig:amati]\nWe first examine the episode 1 as a single GRB. We notice a sharp rise in the luminosity in the first 6 s of emission. We therefore attempted a first interpretation by assuming the first 6 s as the P-GRB component of this independent GRB, as opposed to the remaining 44 s as the extended-afterglow of this GRB. A value of the fit gives *E**t**o**t**e*+*e*− = 3.87 × 1052 ergs and *B* = 1.5 × 10− 4. This would imply a very high value for the Lorentz factor at the transparency of ∼ 5000. In turn, this value would imply a spectrum of the P-GRB peaking around ∼ 300 keV, which is in contrast with the observed temperature of 58 keV. Alternatively, we have attempted a second simulation by assuming all the observed data be part of the extended-afterglow of a GRB, with a P-GRB below the detector threshold. Assuming in this case *E**i**s**o* = *E**t**o**t**e*+*e*−, *B* = 10− 2, and assuming for the P-GRB a duration smaller than 10 s, as confirmed from the observations of all existing P-GRBs, we should obtain an energy of the P-GRB greater than 10− 8 ergs/cm2/s, which should have been easily detectable from Fermi and Swift. Also this second possibility is therefore not viable. We can then conclude generally that in no way we can interpret this episode either as a P-GRB of the second episode, as proved in paragraph 3.2 or, as proved here, as a separate GRB. We then conclude that the fulfillment of the Amati relation does not imply for the source to be necessarily a GRB.\nThe HR variation and the time lag of the two episodes\n-----------------------------------------------------\nWe finally address a further difference between the two episodes, related to the Hardness-Ratio behavior (HR) and their observed time-lag. The first evidence of an evolution of the GRBs power-law slope indexes with time was observed in the BATSE GRB photon spectra. In the context of the fireshell scenario, as recalled earlier, the spectral evolution comes out naturally from the evolution of the comoving temperature, the decrease of the bulk Lorentz Γ factor and from the curvature effect, with theoretically predicted values, in excellent agreement with observations in past GRBs.\nIn order to build the HR ratio, we considered the data from three different instruments: Swift-BAT, Fermi-GBM and the CORONAS-PHOTON-RT-2. The plots obtained with these instruments confirm the existence of a peculiar trend of the hardness behavior: in the first 50 s it is evident a monotonic hard-to-soft behavior, as due to the blackbody evolution of the first episode. For the second episode, the following 50 to 151 s of the emission, there is a soft-to-hard trend in the first 4 s of emission, and a hard-to-soft behavior modulated by the spiky emission in the following 100 s. For the HR ratio we considered the ratio of the count rate detected from a higher energy channel to that of a lower energy channel: HR = ctg(HE)/ctg(LE). In particular, we considered the count rate subtracted for the background, even when this choice provides bad HR data in time region dominated by the background, where the count rate can be zero or negative. For the Swift data, we consider the HR ratio for two different energy subranges: the HR1 ratio shows the ratio of the (50-150 keV) over the (15-50 keV) emission while the HR2 ratio shows the ratio of the (25-50 keV) over the (15-25 keV) emission, see Fig. [fig:no5].\n [fig:no5]\n [fig:no5b]\nA similar trend is found for the Fermi-GBM NaI and RT-2 instruments, see Fig.[fig:no5b]. In particular, the HR from Fermi observations was done considering the counts observed by the b0 BGO detector in the range (260 keV - 40 MeV) and the ones observed by the n4 NaI detector in the range (8 - 260 keV). In Fig. [fig:no5b] it is shown the HR ratio for the Fermi observations, where we rebinned the counts in time intervals of 3 seconds. From this analysis we see that the HR ratio peaks at the beginning of each pulse, also for the second episode pulses, but each peak of the second episode pulses is softer than the previous one, suggesting that these pulses are consequential in the second episode and are in general agreement with the advance of a fireshell in the CBM. Since RT-2 data clearly show both the episodes up to 1 MeV it complements the results obtained by Swift (up to 200 keV) and FERMI (up to 440 keV) in the high and the most interesting energy range. Hardness ratio plot of (250-1000 keV)/(8-250 keV) indicates that first phase of both episodes are the hardest.\nFinally, the evident asymmetry of the first episode, supported by the observations of a large time lag in the high and low energy channels, see fig. [fig:1], suggests a different process at work. There is a very significant softening of the first episode, as reported in, where it is observed a large time lag between the 15-25 keV energy range and the 100-150 keV one: the high energy photons peak ∼ 7 seconds before the photons detected in the 15-25 keV energy range. This large time lag is not observed in the second episode, where the lags are of the order of ∼ 1 s.\nMotivated by these results, we proceed to a most accurate time-resolved spectral analysis of the first episode to identify its physical and astrophysical origin.\nA different emission process in the first episode\n=================================================\nThe time resolved spectra and temperature variation\n---------------------------------------------------\nOne of the most significant outcome of the multi-year work of Felix Ryde and his collaborators, (see e.g. and references therein), has been the identification and the detailed analysis of the thermal plus power-law features observed in a time limited intervals in selected BATSE GRBs. Similar features have been also observed recently in the data acquired by the Fermi satellite. We propose to divide these observations in two broad families. The first family presents a thermal plus power-law(s) feature, with a temperature changing in time following precise power-law behavior. The second family is also characterized by a thermal plus power-law component, but with the blackbody emission generally varying without specific power-law behavior and on shorter time scales. It is our goal to study these features within the fireshell scenario, in order to possibly identify the underlying physical processes. We have already identified in Sec. 4 that the emission of the thermal plus power-law component characterizes the P-GRB emission. We have also emphasized that the P-GRB emission is the most relativistic regime occurring in GRBs, uniquely linked to the process of the black hole formation, see Sec. 5. This process appears to belong to the second family above considered. Our aim here is to see if the first episode of GRB 090618 can lead to the identification of the above first family of events: the ones with temperature changing with time following a power-law behavior on time scales from 1 to 50 s. We have already pointed out in the previous section that the hardness-ratio evolution and the large time lag observed for the first episode points to a distinct origin for the first 50 s of emission, corresponding to the first episode.\nWe have made a detailed time-resolved analysis of the first episode, considering different time bin durations in order to have a good statistic in the spectra and to take into account the sub-structures in the light curve. We have then used two different spectral models to fit the observed data, a classical Band spectrum, and a blackbody with a power-law component.\nIn order to have more accurate constraints on the spectral parameters, we made a joint fit considering the observations from both the n4 NaI and the b0 BGO detectors, covering in this way a wider energy range, from 8 keV to 40 MeV. To avoid some bias due to low photon statistic, we considered an energy upper limit of the value of 10 MeV. We report in the last three columns of the Table [tab:no] also the spectral analysis performed in the energy range of the BATSE LAD instrument (20-1900 keV), as analyzed in, just as a comparison tool with the results described in that paper. Our analysis has been summarized in Figs. [fig:no6], [fig:no17] and in Table [tab:no], where we report the residual ratio diagram as well as the reduced-*χ*2 values for the spectral models considered.\n [fig:no6]\n[tab:no]\n| Time | *α* | *β* | *E*0 (keV) | *χ̃**B**A**N**D*2 | *k**T* (keV) | *γ* | *χ̃**B**B* + *p**o*2 | *k**T**L**A**D* (keV) | *γ**L**A**D* | *χ̃**B**B* + *p**o*, *L**A**D*2 |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| A:0 - 5 | -0.45 ± 0.11 | -2.89 ± 0.78 | 208.9 ± 36.13 | 0.93 | 59.86 ± 2.72 | 1.62 ± 0.07 | 1.07 | 52.52 ± 23.63 | 1.42 ± 0.06 | 0.93 |\n| B:5 - 10 | -0.16 ± 0.17 | -2.34 ± 0.18 | 89.84 ± 17.69 | 1.14 | 37.57 ± 1.76 | 1.56 ± 0.05 | 1.36 | 37.39 ± 2.46 | 1.55 ± 0.06 | 1.27 |\n| C:10 - 17 | -0.74 ± 0.08 | -3.36 ± 1.34 | 149.7 ± 21.1 | 0.98 | 34.90 ± 1.63 | 1.72 ± 0.05 | 1.20 | 36.89 ± 2.40 | 1.75 ± 0.06 | 1.10 |\n| D:17 - 23 | -0.51 ± 0.17 | -2.56 ± 0.26 | 75.57 ± 16.35 | 1.11 | 25.47 ± 1.38 | 1.75 ± 0.06 | 1.19 | 25.70 ± 1.76 | 1.75 ± 0.08 | 1.19 |\n| E:23 - 31 | -0.93 ± 0.13 | unconstr. | 104.7 ± 21.29 | 1.08 | 23.75 ± 1.68 | 1.93 ± 0.10 | 1.13 | 24.45 ± 2.24 | 1.95 ± 0.12 | 1.31 |\n| F:31 - 39 | -1.27 ± 0.28 | -3.20 ± 1.00 | 113.28 ± 64.7 | 1.17 | 18.44 ± 1.46 | 2.77 ± 0.83 | 1.10 | 18.69 ± 1.89 | 4.69 ± 4.2 | 1.08 |\n| G:39 - 49 | -3.62 ± 1.00 | -2.19 ± 0.17 | 57.48 ± 50.0 | 1.15 | 14.03 ± 2.35 | 3.20 ± 1.38 | 1.10 | 14.71 ± 3.52 | 3.06 ± 3.50 | 1.09 |\nWe conclude that both the Band and the proposed blackbody + power-law spectral models fit very well the observed data. Particularly interesting is the clear evolution in the time-resolved spectra, corresponding to the blackbody and power-law component, see Fig. [fig:no6]. In particular the *k**T* parameter of the blackbody presents a strong decay, with a temporal behavior well described by a double broken power-law function, see upper panel in Fig. [fig:no17]. From a fitting procedure we obtain the best fit (R2-statistic = 0.992) for the two decay indexes for the temperature variation are *a**k**T* = -0.33 ± 0.07 and *b**k**T* = -0.57 ± 0.11. In an average value for these parameters on a set of 49 GRBs is given: ⟨*a**k**T*⟩ = -0.07 ± 0.19 and ⟨*b**k**T*⟩ = -0.68 ± 0.24. We note however that in the sample considered in only few bursts shows a break time around 10 s, as it is in our case, see Fig. [fig:no17]. There are two of these bursts, whose analysis presents many similarities with our presented source GRB 090618: GRB 930214 and GRB 990102. These bursts are characterized by a simple FRED pulse, whose total duration is ∼ 40 s, quite close to the one corresponding to the first episode of GRB 090618. The break time *t**b* in these two burst are respectively at 12.9 and 8.1 s, while the decay indexes are *a**k**T* = -0.25 ± 0.02 and *b**k**T* = -0.78 ± 0.04 for GRB 930214 and *a**k**T* = -0.36 ± 0.03 and *b**k**T* = -0.64 ± 0.04 for GRB 990102, see Table 1 in, in very good agreement with the values observed for the first episode of GRB 090618. We conclude that the values we observe in GRB 090618 are very close to the values of these two bursts. We shall return to compare and contrast our results with the other sources considered in, as well as GRB 970828 in a forthcoming publication.\nThe results presented in Figs. [fig:no6],[fig:no17], as well as in Table [tab:no], point to a rapid cooling of the thermal emission with time of the first episode. The evolution of the corresponding power-law spectral component, also, appears to be strictly related to the change of the temperature *k**T*. The power-law *γ* index falls, or softens, with the temperature, see Fig. [fig:no6]. An interesting feature appears to occur at the transition of the two power-law describing the observed decrease of the temperature. The large time lag observed in the first episode and reported in section 6.1 has a clear explanation in the power-law behavior of the temperature and corresponding evolution of the photon index *γ*, Figs. [fig:no6],[fig:no17].\nThe radius of the emitting region\n---------------------------------\nWe turn now to estimate an additional crucial parameter for the identification of the nature of the blackbody component: the radius of the emitter *r**e**m*. We have proved that the first episode is not an independent GRB, not a part of a GRB. We can therefore provide the estimate of the radius of the emitter from non-relativistic considerations, just corrected for the cosmological redshift *z*. We have, in fact, that the temperature of the emitter *T**e**m* = *T**o**b**s*(1 + *z*), and that the luminosity of the emitter, due to the blackbody emission, is\n \n*L* = 4*π**r**e**m*2*σ**T**e**m*4 = 4*π**r**e**m*2*σ**T**o**b**s*4(1 + *z*)4, \nwhere *r**e**m* is the radius of the emitter and *σ* is the Stefan constant. From the luminosity distance definition, we also have that the observed flux *ϕ**o**b**s* is given by:\n \n$$\\label{eq:8.2}\n\\phi\\_{obs} = \\frac{L}{4 \\pi D^2} = \\frac{r\\_{em}^2 \\sigma T\\_{obs}^4 (1+z)^4}{D^2}.$$ \nWe then obtain\n \n$$\\label{eq:radius}\nr\\_{em} = \\left(\\frac{\\phi\\_{obs}}{\\sigma T\\_{ob}^4}\\right)^{1/2} \\frac{D}{(1+z)^2}.$$ \nThe above radius differs from the radius *r**p**h* given in Eq. (1) of and clearly obtained by interpreting the early evolution of GRB 970828 as belonging to the photospheric emission of a GRB and assuming a relativistic expansion with a Lorentz gamma factor Γ:\n \n$$r\\_{ph} = \\hat{\\mathcal{R}} D \\left(\\frac{\\Gamma}{(1.06) (1+z)^2}\\right),$$ \nwhere $\\hat{\\mathcal{R}} = \\left(\\phi\\_{obs}/(\\sigma T\\_{ob}^4)\\right)^{1/2}$ and the prefactor 1.06 arises from the dependence of *r**p**h* on the angle of sight. Typical values of *r**p**h* are at least two orders of magnitude larger than our radius *r**e**m*. We shall return on the analysis of GRB 970828 in a forthcoming paper.\nAssuming a standard cosmological model (*H*0 = 70 km/s/Mpc, Ω*m* = 0.27 and ΩΛ = 0.73) for the estimate of the luminosity distance *D*, and using the values for the observed flux *ϕ**o**b**s* and the temperature *k**T**o**b**s*, we have given in Fig. [fig:no18] the evolution of the radius of the surface emitting the blackbody *r**e**m* as a function of time.\nAssuming an exponential evolution with time *t**δ* of the radius in the comoving frame, we obtain from a fitting procedure the value *δ* = 0.59 ± 0.11, well compatible with *δ* = 0.5. We also notice a steeper behavior for the variation of the radius with time corresponding to the first 10 s, which corresponds to the emission before the break of the double power-law behavior of the temperature. We estimate an average velocity of *v̄* = 4067 ± 918 km/s, R2 = 0.91, in these first 10 s of emission. In episode 1 the observations lead to a core of an initial radius of ∼ 12000 km expanding in the early phase with a sharper initial velocity of ∼ 4000 km/s. The effective Lorentz Γ factor is very low, Γ − 1 ∼ 10− 5.\n| |\n| --- |\n| |\n| |\n| |\n[fig:no17]\n [fig:no18]\nConclusions\n===========\nGRB 090618 is one of the closest (*z* = 0.54) and most energetic (*E**i**s**o* = 2.9 × 1053 ergs ) GRBs up to date. It has been observed simultaneously by the largest number of X and *γ* ray telescopes: Fermi, Swift, AGILE, Konus-WIND, Suzaku-WAM and the CORONAS-PHOTON-RT2. These circumstances have produced an unprecedented set of high quality data as well as the coverage of the instantaneous spectral properties and of the time variability in luminosity of selected bandwidth of the source, see e.g. Figs. [fig:Chak],[fig:1]. In addition there is also the possibility of identifying an underlying supernova event from the optical observations in the light curve of well-defined bumps, as well as from the correspective change in colour after around 10 days from the main event. Unfortunately a spectroscopic confirmation of the presence of such supernova is lacking. We have restricted our attention in this paper to the sole X and *γ* ray emission of the GRB, without addressing the possible supernova component.\nBy applying our analysis within the fireshell scenario, see section 4, we have supported that GRB 090618 is actually composed of two different episodes : episode 1, lasting from 0 to 50 s and episode 2 from 50 s to 151 s after the trigger time. We have also illustrated the recent conclusions presented in, that episode 1 cannot be either a GRB nor a part of a GRB, see section 5. By a time-resolved spectral analysis we have fitted the instantaneous spectra by a blackbody plus an extra power-law component. The temperature of the blackbody appears to have a regular dependence with time, described by two power-law functions: a first power-law with decay index a*k**T* = -0.33 ± 0.07 and the second one with b*k**T* = -0.57 ± 0.11, see Section 7. All these features follow precisely some of the results obtained by Felix Ryde and his collaborators, where the authors analyzed selected temporal episodes in some GRBs observed by BATSE.\nWe have also examined with particular attention, see section 6, the radius *r**e**m* of the blackbody emitter observed in the first episode, given by Eq. ([eq:radius]). We interpret the nature of this episode 1 as originating from what we have defined a proto-black hole, : the collapsing bare core leading to the black hole formation. Within this interpretation, the radius *r**e**m* depends only on the observed energy flux of the blackbody component *ϕ**o**b**s*, the temperature *k**T* as well as on the luminosity distance of the source *D*. We obtained a radius of the emitting region smoothly varying between ∼ 12000 and 70000 km, see Fig. [fig:no18]. Other interpretations associating the origin of this early emission to the GRB main event lead to a different definition for the radius of the blackbody emitter, which results to be larger than our radius by at least two orders of magnitude. We are planning a systematic search for other systems presenting these particular features.\nEpisode 2 is identified as a canonical long GRB which originates from the black hole formation process and lasts in arrival time from 50 s to 151 s after the trigger time. The good quality of data allowed us to search for the P-GRB signature in the early emission of the episode 2. From a detailed analysis we find that the first 4 s of episode 2 are in good agreement with the theoretically predicted P-GRB emission, see section 5.2. The observed spectrum integrated over these 4 s is well fitted by a blackbody with an extra power-law component, where this latter component is mainly due to the early emission of the extended-afterglow, see Fig. [fig:pgrb]. From the temperature observed in the P-GRB, *k**T**P**G**R**B* = 29.22 ± 2.21, and the *E**t**o**t**e*+*e*− energy of the second episode, which we assumed equal to the isotropic equivalent energy of this episode, *E**t**o**t**e*+*e*− = 2.49 × 1053 ergs, we obtained the value of the baryon load of the GRB, see also Fig. [fig:no4], *B* = (1.98 ± 0.15) × 10− 3, and a consequent Lorentz Γ factor at the transparency of Γ∘ = 495 ± 40. We have been able to simulate the temporal and the spectral emission of the second episode, as seen by the Fermi-GBM instrument (8 keV – 10 MeV). As we have shown in Fig. [fig:comp], our simulation succeeds in fitting the light curves as well as the spectral energy distribution emitted in the first main spike of the second episode. The residual emission of the last spikes is reasonably fitted, taking into due account the difficulties in integrating the equations of motion, which after the first interactions of the fireshell with the CBM become hardly predictable. The energetic of the simulation is fulfilled and we find that the emission is due to blobs of matter in the CBM with typical dimensions of *r**b**l* = 1016 cm and average density contrast *δ**n*/*n* ≃ 2 particles/cm3 in an overall average density of 1 particle/cm3. We need to find additional cases of such phenomena to augment our statistic and improve its comprehension.\nParticularly relevant are the first two-dimensional hydrodynamical simulations of the progenitor evolution of a $23\nM\\_{\\sun}$ star close to core-collapse, leading to a naked core, as shown in the recent work of Arnett and Meakin. In that work, pronounced asymmetries and strong dynamical interactions between burning shells are seen: the dynamical behavior proceeds to large amplitudes, enlarging deviations from the spherical symmetry in the burning shells. It is of clear interest to find a possible connection between the proto black hole concept, introduced in this work, with the Arnett and Meakin results: to compare the radius, the temperature and the dynamics of the core we have found in the present work with the naked core obtained by Arnett and Meakin from the thermonuclear evolution of the progenitor star. Particularly relevant is the presence, during this phase of collapse, of strong waves, originated in the mixing of the different element’ shells. Such waves should become compressional, as they propagate inward, but they should also dissipate in non-convective regions, causing heating and slow mixing in these regions of the star. Since the wave heating is faster than radiative diffusion (which is very slow), an expansion phase of the boundary layers will occur, while the iron (Fe) core will contract. There is also the interesting possibility that the CBM clouds observed in GRBs be related to the vigorous dynamics in violent activity of matter ejected in the evolution of the original massive star, well before the formation of the naked core (Arnett D., private communication).\nIt is appropriate to emphasize that these results have no relation with the study of precursors in GRBs done in the current literature. Episode 1 and episode 2 are not temporally separated by a quiescent time. The spectral feature of episode 1 and episode 2 are strikingly different and, moreover, the episode 1 is very energetic, which is quite unusual for a typical precursor event. We finally conclude that for the first time we witness the process of formation of the black hole from the phases just preceding the gravitational collapse all the way up to the GRB emission.\nThere is now evidence that the Proto Black Hole formation has been observed also in other GRB sources. After the submission of this article a second example has been found in GRB 101023, then and a paper about this source was submitted on November 4th 2011 and then published on February 1st 2012. There, extremely novel considerations in the structure of the late phase of the emission in X-ray at times larger than 200 s have been presented in favour of a standard signature in these sources. The possible use of this new family of GRBs as distance indicators is being considered.\nWe thank David Arnett for most fruitful discussions, the participants of the Les Houches workshop “From Nuclei to White Dwarfs and Neutron Stars” held in April 2011 (Eds. A. Mezzacappa and R. Ruffini, World Scientific 2011, in press), as well as the members of the AlbaNova University High Energy Astrophysics group. We are thankful to an anonymous referee for her/his important remarks both on the content and the presentation of our work which have improved the presentation of our paper. LI is especially grateful to Marco Muccino for fruitful discussions about the work concerning this manuscript. We are also greateful to the Swift and Fermi teams for their assistance. One of us, AVP, acknowledges the support for the fellowship awarded for the Erasmus Mundus IRAP PhD program. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.\n141 natexlab#1#1\nAksenov, A. G., Ruffini, R., & Vereshchagin, G. V. 2007, Physical Review Letters, 99, 125003\nAmati, L., Frontera, F., Tavani, M., et al. 2002,, 390, 81\nArnaud, K. 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A., Svensson, R., & Paczynski, B. 1991,, 366, 343\n---\n1. http://heasarc.gsfc.nasa.gov/lheasoft/[↩](#fnref1)\n2. ftp://legacy.gsfc.nasa.gov/swift/data/obs/[↩](#fnref2)\n3. ftp://legacy.gsfc.nasa.gov/fermi/data/gbm/[↩](#fnref3)\n"},"data_id":{"kind":"string","value":"arxiv_0182788"}}},{"rowIdx":45698,"cells":{"content":{"kind":"string","value":"SN 2009E: a faint clone of SN 1987A\n===================================\n### Received XXXXX ; accepted XXXXX\nIn this paper we investigate the properties of SN 2009E, which exploded in a relatively nearby spiral galaxy (NGC 4141) and that is probably the faintest 1987A-like supernova discovered so far. We also attempt to characterize this subgroup of core-collapse supernovae with the help of the literature and present new data for a few additional objects. The lack of early-time observations from professional telescopes is compensated by frequent follow-up observations performed by a number of amateur astronomers. This allows us to reconstruct a well-sampled light curve for SN 2009E. Spectroscopic observations which started about 2 months after the supernova explosion, highlight significant differences between SN 2009E and the prototypical SN 1987A. Modelling the data of SN 2009E allows us to constrain the explosion parameters and the properties of the progenitor star, and compare the inferred estimates with those available for the similar SNe 1987A and 1998A. The light curve of SN 2009E is less luminous than that of SN 1987A and the other members of this class, and the maximum light curve peak is reached at a slightly later epoch than in SN 1987A. Late-time photometric observations suggest that SN 2009E ejected about 0.04 M⊙ of 56Ni, which is the smallest 56Ni mass in our sample of 1987A-like events. Modelling the observations with a radiation hydrodynamics code, we infer for SN 2009E a kinetic plus thermal energy of about 0.6 foe, an initial radius of ∼ 7 × 1012 cm and an ejected mass of ∼ 19 M⊙. The photospheric spectra show a number of narrow (v ≈ 1800 km s− 1) metal lines, with unusually strong Ba II lines. The nebular spectrum displays narrow emission lines of H, Na I, [Ca II] and [O I], with the [O I] feature being relatively strong compared to the [Ca II] doublet. The overall spectroscopic evolution is reminiscent of that of the faint 56Ni-poor type II-plateau supernovae. This suggests that SN 2009E belongs to the low-luminosity, low 56Ni mass, low-energy tail in the distribution of the 1987A-like objects in the same manner as SN 1997D and similar events represent the faint tail in the distribution of physical properties for normal type II-plateau supernovae. \nIntroduction\n============\nThe explosion of supernova (SN) 1987A in the Large Magellanic Cloud (LMC) was an epic event not only because it was the nearest SN in a period of about 4 centuries, but also because it changed significantly the general understanding of the destiny of massive stars. It was commonly believed that hydrogen-rich (H-rich) type II plateau supernovae (SNe IIP) were generated by the explosion of red supergiant (RSG) stars. However, the unusual photometric evolution of SN 1987A, with a broad light curve peak reached about 3 months after the explosion instead of the classical plateau observed in H-rich SNe, and the direct detection of the progenitor star in pre-explosion images changed this general view and proved that the precursor of this SN in the LMC was instead a blue supergiant (BSG, Arnett et al. 1989 and references therein).\nDue to their intrinsic rarity and faint early time luminosity, only very few 1987A-like objects were discovered in the past. Good datasets therefore exist only for a handful of objects (SN 1998A, Woodings et al. 1998, Pastorello et al. 2005; SN 2000cb, Hamuy 2001, Kleiser et al. 2011; SNe 2006V and 2006au, Taddia et al. 2011). In analogy with SN 1987A, some of these objects show broad, delayed light curve peaks in all bands (e.g. SN 1998A), while others (e.g. SN 2000cb and SN 1982F) show 1987A-likeness only in the red bands, while the blue bands display a rather normal type IIP behaviour.\nIn this context, the discovery of a new sub-luminous SN with a light curve comparable to that of SN 1987A, but with other observed properties (fainter intrinsic luminosity at peak, smaller synthesized 56Ni mass and lower expansion velocity of the ejected material) resembling those of some underluminous type IIP SNe (Turatto et al. 1998, Benetti et al. 2001, Pastorello et al. 2004, 2006, 2009) is even more interesting.\nThis paper is organized as follows: in Section [obs] we give basic information on SN 2009E and its host galaxy, and we present photometric and spectroscopic observations of the SN. In Section [constpro] we discuss the properties of the progenitor star and the explosion parameters as derived from the characteristics of the nebular spectrum and by modelling the SN observations. In Section [discussion] we analyse the general properties of the family of SNe similar to SN 1987A, whilst the rate of these events is computed in Section [rate]. Main conclusions are given in Section [summary]. Finally, an Appendix has been included to present our sample of 1987A-like events (Appendix [family]), and to compare the light curves of well monitored objects with those of SN 1987A (Appendix [appB]).\nObservations of SN 2009E\n========================\n [Fig1]\nThe host galaxy\n---------------\nNGC 4141 is classified by HyperLeda[2](#fn2) as an SBc galaxy, rich in H II regions (NED[3](#fn3)). estimated the integrated oxygen abundance to be log(O/H) + 12 = 8.60 (8.74 in the nucleus). They also estimated a star formation rate (SFR) from the integrated flux of H*α*, viz. ∼ 0.6 M⊙ yr− 1.\nAdopting the recessional velocity corrected for Local Group infall into Virgo quoted by HyperLeda (v*V**i**r* = 2158 ± 20 km s− 1) and a Hubble constant H0 = 72 ± 5 km s− 1 Mpc− 1, we obtain a distance of about 29.97 ± 2.10 Mpc (i.e. distance modulus *μ* = 32.38 ± 0.35 mag). The Galactic extinction in the direction of NGC 4141 is very low, i.e. *E*(*B* − *V*) = 0.02 mag. Narrow interstellar Na I *λ**λ*5889,5895 (hereafter Na ID) absorption at the host galaxy rest wavelength is marginally detected in the SN spectra, with an equivalent width (EW) of about 0.12 Å. Adopting the relation between EW*N**a**I**D* and interstellar extinction from, we find a host galaxy reddening of *E*(*B* − *V*) = 0.02 mag[4](#fn4). The total colour excess in the direction of SN 2009E is therefore *E*(*B* − *V*) = 0.04 mag.\nThe discovery of SN 2009E\n-------------------------\nSN 2009E was discovered in the spiral galaxy NGC 4141 on January 3rd, 2009, at an unfiltered magnitude of 17.8. With the distance and reddening estimated in Section [hg], the absolute magnitude at the discovery was about − 14.7. The position of SN 2009E is *α* = 12*h*09*m*49*s*.56 ± 0*s*.03, $\\delta = +58^\\circ50'50\\farcs3 \\pm 0\\farcs1$ (equinox J2000.0), which is $17\\farcs2$ East and $6\\farcs9$ South of the center of the host galaxy. According to, nothing was visible in pre-explosion images obtained on 2008 February 6th (to limiting magnitude of 19.5). We also found pre-explosion archive *g*ʹ, *R**c* and *I**c* images[5](#fn5) obtained on 2008 February 24th at the 0.5-m telescope of the Akeno Observatory/ICRR (Yamanashi, Japan) and no source brighter than *V* = 19.5, *R* = 19.6, *I* = 18.8 mag was visible at the SN position. In this paper, we will adopt January 1.0 UT (*J**D*=2454832.5, see Section [lc]) as an indicative epoch for the core-collapse.\nNGC 4141 also hosted the type II SN 2008X, that exploded $7\\farcs6$ East and $4\\farcs6$ North of the nucleus of the galaxy. Interestingly, SN 2008X was another sub-luminous event. The classification spectrum of SN 2008X obtained a few weeks after the explosion showed that it was a type II SN, although with rather narrow spectral lines, similar to those observed in SN 2005cs.\nProbably the faint apparent magnitude of SN 2009E discouraged astronomers to promptly classify this object. However, later on, noted that SN 2009E brightened by 1 − 1.5 mag during the subsequent two months, and only at about 80 days from the discovery the object was classified by as a type II SN around the end of the H recombination phase. SN 2009E appeared to show some similarity with the peculiar SN 1987A, especially in the late-time light curve brightening and the unusual strength of the Ba II P-Cygni lines. The most obvious differences were in the lower luminosity and the much narrower spectral features.\nLight Curves\n------------\nWhilst professional astronomers initially missed to follow this transient object, amateur astronomers monitored SN 2009E extensively, and it is thanks to their efforts that we can now recover information on the early flux evolution. Most images collected during the period January 2009 to May 2009 were from amateur observations. Since the end of March, we started multiband photometric and spectroscopic follow-up observations with larger, professional telescopes. All photometric data were reduced with standard techniques in IRAF[6](#fn6). Images were first overscan, bias and flat-field corrected. Then, photometric measurements of filtered images were performed using a PSF-fitting technique because suitable template images were not available. However, since the SN location was quite peripheral in the galaxy arm, its luminosity largely exceeded that of the surroundings, except at very late epochs when the SN became weak. We estimated the uncertainty due to the non-flat background by placing in all images several artificial stars close to the SN location. The adopted errors were the r.m.s. of the recovered artificial star PSF magnitudes. Multi-band zero-points for the different nights were computed through observations of standard stellar fields obtained during the same nights as the SN observations. The SN magnitudes were finally fine tuned with reference to a sequence of stars in the field of NGC 4141 (see Figure 1), calibrated by averaging magnitudes measured in selected photometric nights. The magnitudes of the local sequence stars are shown in Table [Tab1], while the final SN magnitudes are reported in Table [Tab2].\n [Fig2]\n [Fig3]\n[Tab1]\n| Star | *U* | *B* | *V* | *R* | *I* |\n| --- | --- | --- | --- | --- | --- |\n| 1 | 18.86 (0.03) | 17.59 (0.02) | 16.52 (0.02) | 15.74 (0.01) | 15.18 (0.01) |\n| 2 | 17.47 (0.02) | 16.88 (0.01) | 16.11 (0.01) | 15.60 (0.01) | 15.20 (0.01) |\n| 3 | 15.85 (0.04) | 15.87 (0.01) | 15.35 (0.01) | 15.01 (0.01) | 14.69 (0.01) |\n| 4 | 18.07 (0.03) | 16.89 (0.01) | 15.41 (0.02) | – | – |\n| 5 | 18.81 (0.03) | 18.73 (0.01) | 18.05 (0.02) | 17.62 (0.01) | 17.19 (0.02) |\n| 6 | 17.75 (0.03) | 17.85 (0.01) | 17.26 (0.01) | 16.83 (0.01) | 16.43 (0.01) |\n| 7 | 20.76 (0.05) | 19.33 (0.02) | 18.17 (0.01) | 17.31 (0.01) | 16.56 (0.02) |\n| 8 | 18.97 (0.05) | 18.91 (0.02) | 18.23 (0.01) | 17.78 (0.01) | 17.37 (0.01) |\n| 9 | – | 19.90 (0.02) | 18.46 (0.02) | 17.47 (0.01) | 16.58 (0.02) |\n| 10 | – | 16.32 (0.01) | 15.70 (0.02) | 15.31 (0.02) | 14.94 (0.01) |\n| 11 | – | 14.45 (0.02) | 13.73 (0.03) | 13.15 (0.04) | 12.83 (0.01) |\n| 12 | – | 16.43 (0.03) | 15.86 (0.03) | 15.48 (0.01) | 15.11 (0.02) |\n| 13 | 16.63 (0.06) | 16.40 (0.01) | 15.67 (0.02) | 15.22 (0.01) | 14.78 (0.02) |\n| 14 | – | 19.86 (0.06) | 18.27 (0.10) | – | – |\n [Fig4]\nSince a number of unfiltered pre-explosion images of NGC 4141 were available, magnitude measurements on the amateurs’ images were performed with the PSF-fitting technique after subtraction of the best seeing host galaxy template image. In this way we minimized the contamination of the background sources near the SN location. The subsequent photometric calibration was performed with the prescriptions discussed in. Since the detectors used by amateur astronomers in their follow-up campaign of SN 2009E have quantum efficiencies preferentially peaking at ∼ 5700 − 6500 Å, unfiltered magnitudes were scaled to match the *R*-band photometry. The SN magnitude errors accounted for the uncertainties in this convertion. The final *R*-band-calibrated magnitudes are reported in Table [Tab2], marked with an asterisk. The *B*, *V*, *R*, *I* light curves of SN 2009E, compared with those of SN 1987A, are shown in Figure [Fig2].\nIn Figure [Fig3] we compare the *B* − *V* and *V* − *I* colour curves of SN 2009E with those of a subsample of 1987A-like events and with the sub-luminous type IIP SN 2005cs. The colour curves of SN 2009E are remarkably similar to those of SN 1987A at all phases. The colour estimates at phase ∼ 25 days (*B* − *V* ≈ 1.2 and *V* − *I* ≈ 0.7 mag) suggest that, in analogy with SN 1987A, SN 2009E became red in a time scale that is shorter than for classical SNe IIP (about 1 month for SN 2009E, while ∼ 2 months for type IIP SNe). Between day 30 and day 110, during the recombination phase, the colours remain almost constant. Then, with the post-peak luminosity decline, the SN becomes redder again, reaching *B* − *V* ≈ 1.4 between 110 and 140 days after explosion (this is remarkably similar to the4 *B* − *V* colour observed in SN 1987A). The *V* − *I* colour, instead, peaks at about 200 days (*V* − *I* ≈ 1.6 mag), which is a bit earlier than observed for SN 1987A (*V* − *I* ≈ 1.4 mag between 230 and 310 days). Subsequently, the colours turn back to the blue again.\nThe *R*-band absolute light curve of SN 2009E is shown in Figure [Fig4], together with those of a number of other type II SNe with a broad, delayed light curve peak, similar to that observed for SN 1987A. In analogy with the early discovery of SN 1987A, also SN 2009E was likely caught soon after the core-collapse. Marginal evidence of the initial sharp peak due to the shock breakout is possibly visible (Figure [Fig4], panel C), although the large error bars of the unfiltered amateur images make this finding rather uncertain.\nNevertheless, we can constrain the explosion epoch of SN 2009E with a small uncertainty through a comparison with the early photometric evolution of SN 1987A. Hereafter we will adopt *J**D*=2454832.5− 5+ 2 as the best estimate for the core-collapse epoch of SN 2009E. The distances and reddenings used for our sample of 1987A-like events shown in Figure [Fig4] are listed in Tables [Tab4] and [Tab5]. SN 2009E is the faintest SN in this sample at all the epochs. In particular, its peak magnitude *M**R* ∼ − 16.2 is about 0.4 mag fainter than that of SN 1987A and its 56Ni mass, as derived from the luminosity of the radioactive tail relative to that of SN 1987A, is 0.040− 0.011+ 0.015 M⊙, the lowest among the objects in our sample.\nAnother remarkable property of SN 2009E is that its *R*-band light curve peaks at about +96d after core-collapse, significantly later than SN 1987A ( ∼ +84 days). This suggests different ejecta parameters for SN 2009E (e.g. smaller initial radius, lower expansion velocity or a more massive envelope, see Section [model]). Interestingly, fainter SNe in our sample seem to reach peak magnitude later than the more luminous ones (see also Appendix [appB]).\nSpectra\n-------\n[Tab3]\n| Date | Average *J**D* | Phase⋆ | Instrumental configuration | Exposure (s) | Resolution (Å)$^\\medbullet$ | Range (Å) |\n| --- | --- | --- | --- | --- | --- | --- |\n| Mar24, 2009 | 2454915.40 | +82.9 | Ekar1.82m+AFOSC+gr.4 | 3600 | 24 | 3490–7790 |\n| Mar25, 2009‡ | 2454915.54 | +83.0 | Asiago1.22m+B&C+g600 | 3600 | 6.1 | 5090–7500 |\n| Apr01, 2009 | 2454922.52 | +90.0 | NOT+ALFOSC+gr4 | 1200 | 13 | 3360-9100 |\n| Apr12, 2009 | 2454934.48 | +102.0 | NOT+ALFOSC+gr4 | 1200 | 16 | 3230-9060 |\n| Apr16, 2009 | 2454938.42 | +105.9 | CAHA2.2m+CAFOS+g200 | 1792 | 9.5 | 3700-10670 |\n| May04, 2009 | 2454955.57 | +123.1 | NOT+ALFOSC+gr4 | 2700 | 13 | 3350-9120 |\n| May13, 2009 | 2454964.62 | +132.1 | TNG+LRS+LRR | 3600 | 12 | 5050-9520 |\n| May20, 2009 | 2454971.55 | +139.1 | TNG+LRS+LRR | 3600 | 12 | 4960-9560 |\n| Jun26, 2009‡ | 2455009.42 | +176.9 | CAHA2.2m+CAFOS+g200 | 2 × 2400 | 9.5 | 4100-9100 |\n| Jul18, 2009 | 2455031.44 | +198.9 | CAHA2.2m+CAFOS+g200+GG495 | 2 × 3600 | 13 | 4800-9690 |\n| Aug15, 2009† | 2455058.84 | +226.3 | CAHA2.2m+CAFOS+g200 | 2 × 3600 | 12 | 3880-10670 |\n| Jan22, 2010 | 2455218.64 | +386.1 | NOT+ALFOSC+gr5 | 3 × 1200 | 14 | 5000–10000 |\n| Jan24, 2010 | 2455220.70 | +388.2 | NOT+ALFOSC+gr4 | 3 × 1200 | 12 | 3550–9080 |\n⋆ Days from explosion (*J**D*=2454832.5); $^\\medbullet$ As estimated from the full width at half maximum of isolated night sky lines. ‡ Poor signal-to-noise spectrum; † Weighted average of 2 spectra obtained on August 14th and 15th.\n [Fig5]\nAs mentioned above, the spectroscopic classification of SN 2009E was performed about 80 days after the SN discovery. To our knowledge, no spectra exist to witness the early time evolution of this object. After the SN classification, we started a regular spectroscopic monitoring covering the SN evolution from near maximum light to the nebular phase. Spectra have been reduced in the IRAF environment using standard prescriptions. After the optimal extraction, the SN spectra were wavelength calibrated using arc lamp spectra and then flux calibrated using flux standard stars observed in the same nights as the SN. A final check with photometry was performed in order to fine tune the flux calibration of the spectra, with a remaining uncertainty in the flux calibration of less than 10%. Basic information on the spectra collected for SN 2009E is listed in Table [Tab3], while the spectral sequence until August 15, 2009 is shown in Figure [Fig5].\nSince the earliest spectrum was obtained about 83 days after the core-collapse, the continuum is quite red, indicative of relatively low temperatures (T*b**b* ≈ 6000 K). All the photospheric spectra show a clear flux deficit at the blue wavelengths (below 4500 Å) that is due to line blanketing from increasingly strong metal lines. These spectra are rich in narrow P-Cygni lines indicating remarkably low velocity of the expanding ejecta. From the position of the minima of the Sc II *λ*6245 and *λ*5527 lines, we measured v*p**h* ≈ 1600 km s− 1 in our earliest spectrum and about 1000 km s− 1 at the end of the photospheric phase (Figure [Fig6]). In the same period, the velocity of the unblended Ba II *λ*6142 line declines from 2200 km s− 1 to 1850 km s− 1.\n [Fig6]\nIn Figure [Fig7], a late time photospheric spectrum of SN 2009E (phase about 3 months after explosion) is compared with spectra of SN 1987A and the 1987A-like event SN 1998A, and with spectra of faint SNe IIP. While the strong Ba II lines are a common feature both for the spectra of SN 1987A and under-luminous, 56Ni-poor type IIP SNe, the narrowness of the spectral lines in SN 2009E is more reminiscent of 1997D-like events.\n [Fig7]\n [Fig8]\nTogether with quite a weak H*α*, we clearly identify prominent lines of Na ID, Ca II, O I, Fe II, Ti II, Sc II and Cr II [see for a detailed line identification for underluminous SNe IIP during the photospheric phase, and for the nebular phase]. The Ba II lines (multiplet 2) are among the strongest spectral features. This is a common characteristic of both SN 1987A and faint SNe IIP. The presence of prominent Ba II lines in the photospheric spectra of SN 1987A was interpreted by as a signature of an over-abundance of Ba in the outer layers of the SN [7](#fn7). However, the presence of prominent Ba II features in underluminous SNe IIP can also be explained invoking temperature effects. The evolution of Na ID is puzzling. The absorption component becomes broader with time (see Figure [Fig5] for the evolution of this spectral feature from 2009 April 16 to May 13). A similar behaviour, although less extreme, is observed also for H*α*. This effect is probably due to line blending, with the Ba II *λ*5854 and *λ*6497 lines (and possibly of other metal features) having an increasing strength relative to Na ID and H*α* respectively. The evolution of the expansion velocities, as deduced from the minima of the P-Cygni of H*α*, Na I, Ba II, Fe II and Sc II, is shown in Figure [Fig6]. Metal lines show a monotonic decline and velocities at the end of the plateau phase (1200 − 1800 km s− 1) that are only marginally higher than those observed in sub-luminous type IIP SNe, whilst H*α* and Na ID display an opposite trend. Since May (around 4 months past explosion, Figure [Fig5]), the classical nebular doublets of [Ca II] *λ**λ* 7291-7323 and [O I] *λ**λ* 6300-6364 become visible, and increase in strength with time.\nA late time spectrum of SN 2009E was obtained at the Nordic Optical Telescope (NOT) about 1 year after explosion (see Figure [Fig8]). Although the transition toward a purely nebular appearence is not complete (e.g. broad P-Cygni features of Na ID, O I *λ*7774 and Ca II NIR are still visible), the spectrum is clearly dominated by prominent H*α* in emission, [O I] *λ**λ*6300,6364 and [Ca II] *λ**λ*7291,7323. In addition, a number of [Fe II] features are detected. In this phase the spectrum of SN 2009E more closely resembles those of SN 1987A and SN 1998A (Figure [Fig8], top), although the spectral lines are significantly narrower than those of other 1987A-like events and are closer to those of sub-luminous, 56Ni-poor SNe (Figure [Fig8], bottom).\nConstraining the progenitor of SN 2009E\n=======================================\nThe ejected oxygen mass\n-----------------------\nThe strength of the [O I] *λ**λ*6300,6364 lines in the nebular spectra of core-collapse SNe can be used to roughly estimate the O mass ejected in the SN explosion. This method has been applied for determining the O masses for a few type IIP SNe. We first make an attempt to estimate the minimum O mass needed to produce the [O I] *λ**λ* 6300-6364 feature with the relation between the O mass and the total flux of the doublet presented by. Using the nebular spectrum of SN 2009E, we measure an observed flux F[*O**I*] ≈ 3.35 × 10− 15 erg s− 1 cm− 2. Accounting for the reddening and distance estimates as mentioned in Section [hg], we obtain for SN 2009E a minimum O mass in the range between 0.05 M⊙ and 0.22 M⊙. As a comparison, derived a minimum O mass range for SN 1987A of 0.3 − 1.1 M⊙. The only other 1987A-like object with nebular spectra available is SN 1998A. Using the spectrum at phase ∼ 344 days (see Figure [Fig8], top), we obtain an O mass in the range 0.24 − 1.01 M⊙.\nIn order to improve the O mass estimate, we adopt the method already used in, which is based on the fact that the luminosity of the [O I] doublet in the nebular phase is known to be powered by the *γ*-rays produced in the radioactive decays of 56Co to 56Fe, and deposited in the O-rich ejecta. Since the mass of O in SN 1987A is reasonably well-known, we can derive the O masses of SNe 2009E and 1998A via comparison with the O mass of SN 1987A. We use the relation linking the O mass with the luminosities of the [O I] doublet[8](#fn8) and of the radioactively powered light curve tail, computed at similar phases :\n \n$$\\label{eq1}\nL\\_{[OI]} = \\eta\\_O \\frac{M\\_O}{M\\_{exc}} L\\_{^{56}Co},$$ \nwhere L[*O**I*] and L56*C**o* are the luminosities of the [O I] doublet and of the radioactive tail (respectively), M*O* is the mass of O, M*e**x**c* represents the whole excited mass in which the radioactive energy is deposited, and *η**O* is the efficiency of the transformation of the deposited energy into [O I] line radiation. Assuming that M*e**x**c* and *η*0 are similar in all 1987A-like events and noting that (since the late light curve of SN 2009E follows the 56Co decay rate)\n \n$$\\frac{L\\_{^{56}Co}(1987A)}{L\\_{^{56}Co}(SN)} \\propto \\frac{M\\_{^{56}Ni}(1987A)}{M\\_{^{56}Ni}(SN)},$$ \nfrom Equation [eq1], we obtain:\n \n$$\\label{eq3} \nM\\_O(SN) = M\\_O(1987A) \\times \\frac{L\\_{[OI]}(SN) \\times M\\_{^{56}Ni}(1987A)}{L\\_{[OI]}(1987A) \\times M\\_{^{56}Ni}(SN)}.$$ \nNow we have all the ingredients needed to estimate the O mass in SNe 2009E and 1998A from Equation [eq3] using the information available for SN 1987A. For SN 2009E we obtain M*O* ≈ 0.60 − 0.75 M⊙ (using as O mass for SN 1987A the extreme values of M*O*, *m**i**n* = 1.2 M⊙ and M*O*, *m**a**x* = 1.5 M⊙, respectively), while for SN 1998A we obtain M*O* ≈ 1.18 − 1.48 M⊙. Therefore, with this method, we determine for the under-luminous SN 2009E an ejected mass of O that is a factor of 2 smaller than that of SN 1987A (note that for SN 2009E also the 56Ni mass is a factor of ∼ 2 smaller). In the case of SN 1998A we obtain an O mass that matches the values of SN 1987A, despite the former object ejected more 56Ni.\nAccording to the models presented by and, a star with main sequence mass of about 18 − 20 M⊙ produces 1.2 − 1.5 M⊙ of oxygen and a 56Ni mass of 0.07-0.1 M⊙. This is in excellent agreement with the values observed in SNe 1987A and 1998A. It is more difficult to provide an interpretation for the abundances displayed by SN 2009E (both M*O* and M56*N**i* are smaller by a factor 2 than those of SN 1987A), although we may tentatively guess that the lower O and 56Ni masses would point toward a lower-mass main sequence star. We remark that the above estimates have been obtained under the simplifying assumption that M*e**x**c* and *η**O* were the same for SNe 1987A, 1998A and 2009E. This is not necessarily true, since we can reasonably expect a range of values for these parameters in 1987A-like events, and this would significantly affect our O mass estimates.\nModelling the data of SN 2009E\n------------------------------\nOther physical properties of SN 2009E (namely, the ejected mass, the progenitor initial radius and the explosion energy) have been estimated by performing a model vs. data comparison using two codes that compute the bolometric light curve, the evolution of the ejecta velocity and the continuum temperature at the photosphere of a SN. The first code, based on a simplified semi-analytic treatment of the ejecta evolution, is used to perform a preparatory study in order to constrain the parameter space. The second code, that includes a more accurate treatment of radiative transfer and radiation hydrodynamics, computes the tighter grid of models needed for the final comparison. This is similar to the procedure adopted for SN 2007od in.\n [Fig9]\nThe bolometric light curve was computed from the observed multicolor light curves using the assumptions on extinction and distance modulus reported in Section [hg]. We have adopted the explosion epoch estimated in Section [lc] (*J**D* = 2454832.5− 5+ 2). Finally, we have assumed that SN 2009E had the same color evolution as SN 1987A (this is a reasonable assumption, see e.g. Figure [Fig3]), and hence\n \n$$L(2009E) = L(1987A) \\times \\frac{L\\_{BVRI}(2009E)}{L\\_{BVRI}(1987A)},$$ \nwhere *L* and *L**B**V**R**I* are the bolometric and quasi-bolometric (*B**V**R**I*) luminosities of the two SNe.\nThe best *χ*2 fit to the bolometric light curve of SN 2009E obtained with the full radiation-hydrodynamics code returned values of a total (kinetic plus thermal) energy (E) of 0.6 foe, an initial radius (R0) of 7 × 1012 cm, and an envelope mass (M*e**j*) of 19*M*⊙ (see Figure [Fig9]). As a comparison, the fit with the semi-analytic code gave E = 1.3 foe, R0 = 6 × 1012 cm, and M*e**j* = 26 M⊙ (see again Figure [Fig9]). The parameters inferred from the two best-fits are in reasonable agreement. The envelope mass (and consequently the total energy) are slightly overestimated by the semi-analytic code as a consequence of the different mass distribution. Indeed, given the assumption of uniform density throughout the ejecta, the semi-analytic code tends to gather more mass (and hence more energy) in the external, faster-moving layers. The 56Ni distribution also affects the mass estimate because, as a consequence of the simplified uniform distribution adopted in the semi-analytic code, less energy is released at the end of the photospheric phase and, hence, a more massive envelope is needed to sustain the recombination phase.\nThe ejected 56Ni mass of 0.039 M⊙ that is adopted in the semi-analytic code is in an excellent agreement with that estimated in Section [lc], whilst the one inferred from the radiation-hydrodynamic code is slightly higher (0.043 M⊙). Since the latter code accounts also for the amount of material which (eventually) falls back onto the central remnant during the post-explosive evolution, the numerical simulations need a larger initial 56Ni mass to reproduce the observed late-time light curve of SN 2009E.\nIn Figure [Fig9] the evolutions of the photospheric velocity and temperature are also shown. The agreement between models and observations in the late photospheric phase is good, with possibly some discrepancy in matching the photospheric velocities at later times ($\\gtrsim$ 4 months). The decline of the model velocity profile is faster than that inferred from the observations, probably because some extra radioactive decay heating occurs in the inner part of the ejecta of SN 2009E as a consequence of a different (i.e. more centrally condensed) distribution of 56Ni.\n[Tab4]\n| SN | Host galaxy | Type1 | *μ*2 | A*B*, *M**W* | *M**B* | d*S**N*0/r25 | P04 | P04 corr. |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| 1909A | NGC 5457 | SABc | 29.343 | 0.037 | -21.01 | 1.119 | 8.59 | 8.3010 |\n| 1982F | NGC 4490 | SBcd | 29.92 | 0.093 | -20.25 | 0.23 | 8.53 | 8.6011 |\n| 1987A | LMC | SBm | 18.504 | 0.324 | -18.02 | - | 8.31 | - |\n| 1998A | IC 2627 | SABc | 32.19 | 0.518 | -20.03 | 0.66 | 8.51 | 8.41 |\n| 1998bt | A132541-2646 | - | 36.415 | 0.244 | -13.25 | - | 7.98 | - |\n| 2000cb | IC 1158 | SABc | 32.686 | 0.491 | -19.34 | 0.93 | 8.46 | 8.24 |\n| NOOS-005 | 2MASX J05553978-6855381 | S | 35.46 | 0.6568 | -20.01 | 1.26 | 8.51 | 8.17 |\n| 2004em | IC 1303 | Sc | 34.07 | 0.466 | -19.73 | 0.96 | 8.49 | 8.26 |\n| 2005ci | NGC 5682 | Sb | 32.73 | 0.141 | -18.07 | 0.66 | 8.36 | 8.26 |\n| 2006V | UGC 6510 | SABc | 34.367 | 0.125 | -20.97 | 1.31 | 8.59 | 8.22 |\n| 2006au | UGC 11057 | Sc | 33.387 | 0.742 | -20.64 | 0.90 | 8.56 | 8.36 |\n| 2009E | NGC 4141 | SBc | 32.38 | 0.086 | -17.75 | 0.68 | 8.33 | 8.22 |\n1 Morphologic type as quoted by HyperLeda. \n2 Unless otherwise specified, distance moduli are computed from HyperLeda’s v*V**i**r*, adopting H0 = 72 km s− 1 Mpc− 1. \n3 Distance computed from the tip of the red giant branch. Cepheid distances are found to be too sensitive to metallicity. The metallicity correction is even more crucial in the case of NGC 5457 where inner metal-poor and outer metal-rich cepheids give uncorrected distance moduli that may differ by 0.3-0.4 mags. \n4 For LMC we adopted a cepheid distance, well consistent with that derived from the TRGB. \n5 SN 1998bt was discovered in the course of the Mount Stromlo Abell cluster supernova search. Unfortunately, no spectroscopic classification exists for this SN. \nAccording to, there is a very faint host galaxy at the position of SN 1998bt, but no spectrum was ever obtained of this galaxy. We assume that the galaxy hosting SN 1998bt belongs to a cluster monitored by the search whose average redshift is z = 0.046. \n6 Average EPM distance from Table 4.1 of. \n7 Distance as in, but obtained adopting an Hubble Constant H0 = 72 km s− 1 Mpc− 1. \n8 This galaxy is behind LMC, so the extinction reported here includes both Galaxy and LMC contributions (see text for references). \n9 Computed using the position angle provided by. \n10 Other sources give 12+log(O/H) ≈ 7.7-7.8 at the position of SN 1909A. \n11 An alternative estimate from gives 12+log(O/H) = 8.35. \n[Tab5]\nccccccccccc SN & E*t**o**t*(B-V)*a* & *J**D**e**x**p**l*. & Band & *J**D**p**e**a**k* & t-t0 (days) & *m**p**e**a**k*(filter) & *M**p**e**a**k* & M(56Ni) & Sources \n1909A & 0.009 & 2418310 ± 30 & *B* & 2418383 ± 7 & 73 ± 31 & 14.47 ± 0.30 & -15.91 & 0.13M⊙*b* & 1 \n1982F & 0.022 & 2445014 ± 15 & *B* & 2445087 ± 4 & 73 ± 16 & 16.40 ± 0.02 & -13.61 & – & 2,3,4 \n& & & *V* & 2445097 ± 3 & 83 ± 15 & 14.95 ± 0.10 & -15.04 & & \n1987A & 0.19 & 2446849.816*c* & *B* & 2446932.0 ± 1.1 & 82.2 ± 1.1 & 4.52 ± 0.01 & -14.77 & 0.075M⊙ & 5 \n& & & *V* & 2446932.2 ± 1.0 & 82.4 ± 1.0 & 2.96 ± 0.01 & -16.13 & & \n& & & *R* & 2446933.6 ± 1.0 & 83.8 ± 1.0 & 2.27 ± 0.01 & -16.67 & & \n& & & *I* & 2446935.2 ± 0.9 & 85.4 ± 0.9 & 1.90 ± 0.01 & -16.88 & & \n1998A & 0.120 & 2450801 ± 4 & *B* & 2450876.3 ± 7.4 & 75.3 ± 8.4 & 17.59 ± 0.13 & -15.12 & 0.09M⊙ & 6 \n& & & *V* & 2450885.4 ± 8.3 & 84.4 ± 9.2 & 16.31 ± 0.09 & -16.28 & & \n& & & *R* & 2450885.6 ± 3.9 & 84.6 ± 5.6 & 15.73 ± 0.06 & -16.78 & & \n& & & *I* & 2450889.2 ± 5.5 & 88.2 ± 6.8 & 15.37 ± 0.07 & -17.05 & & \n1998bt & 0.057 & 2450831 ± 30 & *V* & 2450905 ± 6 & 74 ± 31 & 19.81 ± 0.05 & -16.79 & & 7,0 \n& & & *R* & 2450908 ± 8 & 77 ± 31 & 19.38 ± 0.08 & -17.18 & & 8,0 \n2000cb& 0.112 & 2451653.8 ± 1.4 & *B**d* & & & 18.23 ± 0.02 & -14.94 & 0.11M⊙ & 8,9,10 \n& & & *V* & 2451714.6 ± 4.7 & 60.8 ± 4.9 & 16.58 ± 0.02 & -16.47 & & \n& & & *R* & 2451725.2 ± 2.1 & 71.4 ± 2.5 & 15.95 ± 0.02 & -17.03 & & \n& & & *I* & 2451729.0 ± 1.4 & 75.2 ± 2.0 & 15.56 ± 0.02 & -17.34 & & \nNOOS-005&0.160 & 2452855 ± 15 & *I* & 2452939.5 ± 2.7 & 84.5 ± 15.2 & 18.22 ± 0.02 & -17.51 & 0.13M⊙ & 0 \n2005ci & 0.033? & 2453521 ± 3 & *u**n**f*. & & & < 17.5 & < -15.3 & & 10 \n2006V*e* & 0.029 & 2453748 ± 4 & *B* & 2453823.7 & 75.7 & 18.29 ± 0.01 & -16.20 & 0.13M⊙ & 11 \n2006au*e* & 0.312 & 2453794 ± 9 & *B* & 2453865.5 & 71.5 & 18.63 ± 0.02 & -16.06 & < 0.07M⊙ & 11 \n2009E & 0.04 & 2454832.5 ± 2.0 & *B* & 2454919.0 ± 6.5 & 86.5 ± 6.8 & 18.01 ± 0.03 & -14.54 & 0.04M⊙ & 0 \n& & & *V* & 2454924.9 ± 5.9 & 92.4 ± 6.2 & 16.76 ± 0.04 & -15.74 & & \n& & & *R* & 2454928.3 ± 2.8 & 95.8 ± 3.4 & 16.25 ± 0.02 & -16.21 & & \n& & & *I* & 2454928.3 ± 6.0 & 95.8 ± 6.3 & 15.98 ± 0.03 & -16.46 & & \n*a* The SN extinction is computed accounting for both Milky Way and host galaxy reddening components; in the case of NOOS-005, since the host galaxy is projected behind LMC, we also accounted for the contribution of LMC. \n*b* Computed using the B-band detections at days +429 and +432, and assuming the same colour evolution as SN 1987A. Using the detection at day +216, we would obtain M(56Ni) ≈ 0.35M⊙. \n*c* and references therein. \n*d* The broad delayed maximum typical of 1987A-like SNe is not visible in the B band; here we report the magnitude of the pseudo-plateau visible after the early time B-band maximum. \n*e* For a comprehensive information on the multi-band light curve parameters, see Table 7 of. \n0 = This paper; 1 = Sandage & Tammann 1974; 2 = Yamagata & Iye 1982; 3 = Tsvetkov 1984; 4 = Tsvetkov 1988; 5 = Whitelock et al. 1989 (and references therein); 6 = Pastorello et al. 2005; 7 = Germany et al. 2004; 8 = Hamuy 2001; 9 = Hamuy & Pinto 2002; 10 = Kleiser et al. 2011; 11 = Taddia et al. 2011.\nFinally, adopting a mass of the compact remnant of about 2 M⊙ (and assuming a negligible pre-SN mass loss), we derive a final mass of 21 M⊙ for the progenitor of SN 2009E, which is slightly higher than our rough estimate obtained throught the O mass estimate (Section [OI]).\nSN 2009E and other 1987A-like events\n====================================\nAlthough 1987A-likeness was claimed for a number of objects in the past, reasonable observational evidence exists for only a small number of events [9](#fn9). Additionally, for a few of them [SN 2004em, ; SN 2005ci, ] data are not published yet or are incomplete. Our limited 1987A-like sample includes the objects that are briefly described in Appendix [family] (and whose light curves are shown in Appendix [appB]). The sample has been selected among objects that are spectroscopically classified as type II (without showing type IIn-like spectral features), and with light curves showing a slow rise to maximum.\nMain parameters of the host galaxies of our sample of 1987A-like events are listed in Table [Tab4]. Remarkably, most SNe in our sample occured in late-type galaxies (Scd or irregular). In addition, some of them were hosted by intrinsically faint (dwarf) galaxies, which are thought to have low-metallicities. However, at least half of the objects of our sample occurred in luminous spirals. Although luminous spirals are believed to be rather metal-rich, the 1987A-like events hosted in such galaxies occurred in peripheral -and hence more metal deficient- regions of the galaxies. Environments with low-to-moderate metallicity (slightly sub-solar) appear to be a fairly common characteristic for 1987A-like objects.\nIn Table [Tab5] we report the values of some observational parameters for our SN sample. The magnitudes of the broad peaks seem to be correlated with the ejected 56Ni masses (i.e. the objects with the brightest peak magnitudes are more 56Ni-rich). The 56Ni masses reported in Table [Tab5] have been computed through a comparison between the late-time *uvoir* light curves of the different SNe with that of SN 1987A. The bolometric corrections were computed with reference to SN 1987A. The derived 56Ni masses span a range between about 0.04 M⊙ and 0.13 M⊙[10](#fn10), similar to that observed in normal type IIP SNe, although cases of extremely 56Ni-poor 1987A-like events have not been found so far.\nUnfortunately, the sub-sample of 1987A-like SNe with an extensive data sets necessary for modelling is very small. This is necessary to constrain the properties of the progenitor stars. The rather large ejected mass, the moderate amount of synthesized 56Ni, and the small initial radius found for SN 2009E (Section [model]) fit reasonably well within the framework of a relatively compact and massive progenitor, and are in a good agreement with the results found in previous works on 1987A-like events. Using an older version of the semi-analytic modelling code than the one adopted here, obtained the following estimates for SNe 1987A and 1998A: E(1987A) = 1.6 foe, E(1998A) = 5.6 foe; M*e**j*(1987A) = 18 M⊙, M*e**j*(1998A) = 22 M⊙; M56*N**i*(1987A) = 0.075 M⊙, M56*N**i* = 0.11 M⊙; R0(1987A) $\\gtrsim 5 \\times 10^{12}$ cm, R0(1998A) $\\gtrsim 6 \\times 10^{12}$ cm. The major difference lies in the ejected 56Ni masses of SNe 1987A and 1998A, which are × 2 and × 3 larger (respectively) than the one we have derived for SN 2009E (Section [model]). Explosion parameters for a few additional 1987A-like SNe are also available in the literature. Using a one-dimensional Lagrangian radiation-hydrodynamics code, derived the following parameters for the luminous 1987A-like SN 2000cb: E(2000cb) = 2 foe, M*e**j*(2000cb) = 17.5 M⊙, M56*N**i*(2000cb) = 0.10 M⊙ and R0(2000cb) = 3 × 1012 cm. These values are slightly different from those derived by by modelling the spectroscopic and photometric data of SN 2000cb with a different hydrodynamic code. They inferred a presupernova radius of 2.4 × 1012 cm, an ejected mass of 22.3 M⊙, an energy of 4.4 foe, and a mass of radioactive 56Ni of 0.083 M⊙[11](#fn11). With a semi-analytic approach, estimated relevant physical parameters for two 1987A-like events, SNe 2006V and 2006au: E(2006V) = 2.4 foe, E(2006au) = 3.2 foe; M*e**j*(2006V) = 17.0 M⊙, M*e**j*(2006au) = 19.3 M⊙; M56*N**i*(2006V) = 0.127 M⊙, M56*N**i*(2006au) ≤ 0.073 M⊙; R0(2006V) = 5.2 × 1012 cm, R0(2006au) = 6.3 × 1012 cm.\nIt is evident that 1987A-like SN explosions have similar kinetic energies (a few × 1051 erg) and ejected masses (17 to 22 M⊙), whilst they appear to span a factor of 3 in 56Ni masses. It is intriguing to note that the modelling reported above suggest that *all the 1987A-like SNe would be produced by the explosions of BSG progenitors that have initial radii ranging between 35 and 90 R⊙, and with final masses around 20 M⊙*. These progenitor masses are systematically higher than those estimated for the RSG progenitors of classical SNe IIP.\nRate of 1987A-like events\n=========================\nWhile it would be of interest to know the relative rate of 1987A-like events vs. normal type II SNe, the heterogeneous properties of this class and the uncertainties in the sample population from which the events reported in Table [Tab4] are extracted prevent us from providing any accurate estimates. With the aim to derive an educated guess, the completeness of the SN searches with respect to the 1987A-like objects list can be assessed by comparing the distance modulus distribution of 1987A-like events with that of a suitable reference sample. For the latter, we extracted from the Asiago Supernova Catalogue the list of all type II SNe. If we limit the distance modulus to about 35 mag (at higher redshift incompleteness appears to be significant) it seems that 1987A-like events would account for less than 1.5% of all type II SNe. This value, that can be considered as a lower limit for the relative rate of 1987A-like events, is very similar to that derived by based on the two 1987A-like SNe discovered by the LOSS.\nHowever, the identification of 1987A-like events demands a fairly good light curve coverage that is not available for many of the type II SNe listed in the catalogue. Actually, reasonably good light curves of type II (IIP, IIL, IIb, IIn) SNe exist for about 300 objects. This approximate estimate was obtained including both SNe whose data have been published in literature, and SNe spectroscopically classified whose photometric follow-up information is available in public pages of current SN monitoring programs[12](#fn12). From this estimate, we infer that 1987A-like events comprise about 4% of all type II SNe. This has to be considered as a sort of upper limit, since we expect that the astronomical community has been investing more efforts to follow peculiar 1987A-like SNe rather than other type II SN sub-types because of their intrinsical rarity. Therefore, we estimate that SNe photometrically similar to SN 1987A comprise between ∼ 1.4 and 4% of all type II SNe, corresponding to about 1-3% of all core-collapse SNe in a volume-limited sample.\nSummary\n=======\nNew data for the peculiar type II SN 2009E have been presented. Its light curve is very similar to that of SN 1987A, with a few main differences: the light curve peak is shifted in phase by about +10 days with respect to that of SN 1987A, its luminosity is fainter than that of SN 1987A and the amount of 56Ni deduced from the luminosity of the radioactive tail is about 0.04M⊙. This is the lowest ejected 56Ni mass estimated for any 1987A-like event studied so far. The spectra of SN 2009E appear to be different from those of SN 1987A (and similar events), while they share major similarities with those of sub-luminous, 56Ni-poor SNe IIP (e.g. SN 2005cs) in terms of line velocities and relative line strengths.\nIn analogy with what has been deduced for SN 1987A, also the observed properties of SN 2009E are consistent with those expected from the explosion of a BSG star. Modelling the data of SN 2009E, we obtained a relatively under-energetic explosion (*E* = 0.6 foe), an initial radius of of 7 × 1012 cm and an ejected mass of 19*M*⊙, pointing toward a relatively high-mass precursor star (21*M*⊙), very similar to the BSG progenitor inferred for SN 1987A.\nFinally, we compared the observed properties of a dozen of 1987A-like SNe available in the literature, and found that this subgroup of type II SNe shows significant variety in the explosion parameters and in the characteristics of the host galaxies (although they seem to prefer moderately metal-poor environments). This would suggest for these objects a distribution in the space of parameters which is similar to that observed in more classical type IIP SNe, although they are intrinsically rare ($\\lesssim$ 3% of all core-collapse SNe in a volume-limited sample).\nThis work is based in part on observations collected at the Italian 3.58-m Telescopio Nazionale Galileo (TNG), the 2.56-m Nordic Optical Telescope, the 2.2-m Telescope of the Centro Astronómico Hispano Alemàn (Calar Alto, Spain) and the 1.82-m Copernico Telescope on Cima Ekar (Asiago, Italy). The TNG is operated by the Fundación Galileo Galilei of the Instituto Nazionale di Astrofisica at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. We thank the support astronomers at the TNG, the 2.2-m Telescope at Calar Alto and the Nordic Optical Telescope for performing the follow-up observations of SN 2009E. SB, FB, EC, MT and AH are partially supported by the PRIN-INAF 2009 with the project ‘Supernovae Variety and Nucleosynthesis Yields’. MLP acknowledges financial support from the Bonino-Pulejo Foundation. SM acknowledges financial support from the Academy of Finland (project: 8120503). We are grateful to the TriGrid VL project and the INAF-Astronomical Observatory of Padua for the use of computer facilities.\nEP acknowledges the project Skylive of the Unione Astrofili Italiani (UAI) and the Skylive Remote Facility - Osservatorio B40 Skylive, Catania, Italy (*http://www.skylive.it*). We thank the amateur astronomers of the Associazione Astronomica Isaac Newton (*http://www.isaacnewton.it*) and of the Associazione Astronomica Cortina (*http://www.cortinastelle.it/*) for providing images of SN 2009E obtained during the SMMSS and CROSS supernova search programs: G. Iacopini (Osservatorio di Tavolaia, S. Maria a Monte, Pistoia), R. Mancini and F. Briganti (Stazione di Osservazione di Gavena, Cerreto Guidi, Italy), A. Dimai (Osservatorio di Col Driuscé, Cortina, Italy). We also thank J.M. Llapasset, O. Trondal, Boyd, for sharing their images with us, I. Bano and A. Englaro (Osservatorio Astronomico Polse di Cougnes) for their help with observations, A. Siviero for allowing ToO observations of SN 2009E at the 1.82-m Telescope of Asiago (Mt. Ekar, Italy) and T. Iijima for providing us the B&C spectrum obtained at the 1.22-m Galileo Telescope (Osservatorio Astrofisico di Asiago). AP is grateful to Mario Hamuy, Io Kleiser, Dovi Poznanski and Lisa Germany for sharing their data of SNe 2000cb and 1998bt.\nThis manuscript made wide use of information contained in the Bright Supernova web pages (maintained by D. Bishop), as part of the Rochester Academy of Sciences (*http://www.RochesterAstronomy.org/snimages/*). We also used information collected from the the web site SNAude des Supernovae (*http://www.astrosurf.com/snaude/*). 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K. 2004, AJ, 128, 1606\nBasic information on the sample of SN 1987-like transients\n==========================================================\n1. **SN 1987A** is the prototype of this family of H-rich core-collapse SNe, and -since it exploded in the nearby Large Magellanic Cloud- it is the best studied SN ever. The SN was discovered on 1987 February 24th by Shelton. Strong constraints on the time of the core-collapse came from the detection of a neutrino burst on February 23.316 UT by IMB and Kamiokande II. Extensive data sets at all wavelengths are provided by a large number of publications. More recent papers have unveiled the complex structure of the circumstellar environment of SN 1987A and its interaction with the SN ejecta. SN 1987A is one of the few objects for which we have robust contraints on the nature of the progenitor star. Pre-explosion images of the source at the SN position (Sk − 69 202) showed it to be a blue (B3 I) supergiant whose mass is estimated to be around 20 M⊙. A small initial radius of the progenitor star is also used to explain the unusual, slow rise to maximum of the light curves of SN 1987A. In this paper we adopt as distance modulus of LMC and as total reddening toward SN 1987A the values of *μ* = 18.50 mag and *E*(*B* − *V*) = 0.19 mag.\n2. **SN 1909A** is a historical SN that exploded in M101. Its photometric evolution shows striking similarity with that of SN 1987A. A well-sampled light curve from photographic plates rescaled to the B-band system was published by. We assume negligible extinction toward SN 1909A and a distance modulus of *μ* = 29.34 mag. Although multiband observations are missing for this event, the B-band absolute magnitude at maximum (*M**B* ≈ − 15.9, see Table [Tab5]) suggests that SN 1909A is one of the intrinsically brightest SNe in our sample.\n3. **SN 1982F** in NGC 4490 is a poorly followed under-luminous type II SN. Sparse data around maximum from photographic plates were published by and. The light curves share remarkable similarity with SN 2000cb (see below), more than with SN 1987A. No spectrum exists for this object to our knowledge.\n4. **SN 1998A** is a well studied 1987A-like event which exploded in a spiral arm of the SBc galaxy IC 2627. The SN was discovered on 1998 January 6 by the Automated Supernova Search Program of the Perth Astronomy Research Group. Optical photometry and spectroscopy were published by and. Adopting the HyperLeda recessional velocity corrected for local infall onto Virgo v*V**i**r* = 1976 km s− 1, we obtain a distance of 27.4 Mpc (*μ* = 32.19 mag). Since there was no evidence of additional extinction in the host galaxy, we adopt the same total reddening estimate as in, i.e. *E*(*B* − *V*) = 0.12 mag.\n5. **SN 1998bt** was discovered in the Abell cluster 1736 by the Mount Stromlo Abell cluster supernova search team on March 10, 1998. Initially, no background galaxy was seen to be associated with the SN. However, subsequent deep imaging of the SN field revealed a very faint parent galaxy of *R* = 23.4. Unfortunately, spectroscopic classification for this SN does not exist. However, the overall behaviour of the light curve (see Figure [FigA1]) is reminiscent of that of SN 1987A.\n6. **SN 2000cb** in IC 1158 is one of the best followed 1987A-like events in our sample. Discovered on April 27.4 2000 using the 0.8-m Katzman Automatic Imaging Telescope, it was classified as a young type II SN by and. Optical photometric and spectroscopic observations have been presented by and. The photometric evolution of SN 2000cb is different from that of SN 1987A. The light curve in the *B* band shows a maximum at ≈ 40 days past core-collapse, followed by a slow decline up to 60 days and a short pseudo-plateau. At about 90 days a steep post plateau decline to the radioactive tail is visible, as in normal type IIP SNe. The evolution in the *V**R**I* bands is somewhat different, showing broad light curve peaks at 60 − 80 days (depending on the band) after core-collapse. The light curve peaks are significantly broader than those of SN 1987A. We adopted the observed parameters of SN 2000cb as presented in. The explosion epoch, the adopted distance and the total reddening are those of. Their values are listed in Tables [Tab4] and [Tab5].\n7. **OGLE-2003-NOOS-005** never had a SN designation. It was discovered by the Optical Gravitational Lensing Experiment collaboration at *α* = 5*h*55*m*38*s*.07 and $\\delta = -68^\\circ55'47\\farcs1$ (J2000.0). It exploded in a faint spiral galaxy labelled by the NED database as 2MASX J05553978-6855381, behind LMC, and falls in the OGLE field LMC198.6 9. The well-followed I-band light curve of OGLE-2003-NOOS-005[13](#fn13) well matches that of SN 1987A. Unfortunately multiband observations for this SN do not exist. The redshift of the host galaxy was spectroscopically determined measuring the positions of a few selected spectral features of a host galaxy spectrum obtained on 2009 August 14 with NTT (equipped with EFOSC2 and grism 11; resolution = 22 Å). Despite the low signal-to-noise, we identified a few absorption features (Ca II H&K, the *g*-band *λ*4200, Mg Ib *λ*5173), and H*α* in emission. In addition the emission line of [O II] *λ*3727 was marginally detected. This allowed to fix the redshift at z = 0.0302 ± 0.0008 (v*r**e**c* ≈ 9151 ± 236 km s− 1), which corresponds to a distance of about 127 Mpc (*μ* = 35.52 mag). These values are remarkably similar to those reported by NED. and HyperLeda (v*r**e**c* ≈ 9177 ± 60 km s− 1). The recessional velocity corrected for Local Group infall into the Virgo Cluster is slightly lower (v*V**i**r* ≈ 8882 km *s*− 1). Adopting the Hubble Constant value of H0 = 72 km s− 1 Mpc− 1, we obtain a distance of 123.4 Mpc, i.e. distance modulus *μ* = 35.46 mag. We also assume in our analysis a total reddening toward the transient of *E*(*B* − *V*) = 0.16 ± 0.10 mag. This was determined including the contribution of the Galaxy *E**M**W*(*B* − *V*) = 0.075 mag and the reddening at the position of 2MASX J05553978-6855381 due to the intrinsic contribution of LMC that was computed using the maps of (*E**L**M**C*(*B* − *V*) = 0.087 mag). The I-band light curve of this SN is shown in Appendix B.\n8. **SN 2004em** was discovered by on Sept. 14, 2004 when the SN was at *m**a**g* = 17.5. The transient, later classified as a young type II by, was hosted in an Sc galaxy (IC 1303). The object was significantly reddened by Galactic dust. Preliminary information on the Caltech Core-Collapse Project light curve has been provided by.\n9. **SN 2005ci**, a LOSS discovery, was classified as a type II SN by on the basis of the presence of P-Cygni Balmer lines. Its peculiar light curve was first noted by, and some early-time unfiltered photometry was provided by, showing a clear light curve rise by about 2.5 mag in ∼ 45 days. Unfortunately, with the modest amount of data available so far, we cannot estimate the main SN parameters. The host galaxy, NGC 5682, is an Sb-type; the Galaxy reddening is modest, *E*(*B* − *V*) = 0.033 mag, but the relatively red continuum shown by the spectrum presented by (see their Figure 12) suggests a non negligible host galaxy reddening. The narrow host galaxy Na ID feature is not clearly detected and the amount of internal reddening cannot be constrained.\n10. **SN 2006V** was discovered in the course of the Taiwan Supernova Survey on Feb. 4, 2006 UT at a magnitude of about 18. The object was classified as a type II SN after maximum by and extensively followed by the Carnegie Supernova Project collaboration, who noted the slowly rising, 1987A-like light curve. The galaxy hosting SN 2006V is UGC 6510, which is classified as a face-on spiral (SABc, HyperLeda source) suffering of small Galactic reddening. The lack of detection of narrow Na ID absorption at the redshift of UGC 6510, suggest no additional host galaxy extinction toward SN 2006V. A comparison between the multi-band light curves of SNe 2006V and 1987A can be found in Figure 3 of.\n11. **SN 2006au** was a discovery of the Tenagra Observatory Supernova Search using the 0.35-m Tenagra telescope in Oslo (on Mar. 7.20 UT). At the discovery, the object had *m**a**g* 17.2. SN 2006au was later classified by the Nearby Supernova Factory as a type II SN. Again, a comprehensive study is presented in and comparisons with the light curves of SN 1987A are shown in their Figure 4. The host galaxy, UGC 11057, is a late spiral (possibly an Sc-type, according to HyperLeda) with rather large Galactic reddening, i.e. *E*(*B* − *V*) = 0.172 mag. However, found that SN 2006au suffered also of large extinction due to dust in the host galaxy, deriving a total reddening of *E*(*B* − *V*)*t**o**t* = 0.312 mag.\nLight Curves of 1987A-like events\n=================================\n[FigA1]\nIn this section, the light curves of eight 1987A-like type II SNe are compared with those of the prototype SN 1987A (solid or dashed blue lines, see Figures [FigA1] and [FigA2]). The light curves of SN 1987A are arbitrarily shifted in magnitude to match the peak magnitudes of the other objects. In the comparison, we assumed the *J**D*s listed in Table [Tab4] as explosion epochs. For a few individual events (e.g. SNe 1982F or 2000cb) the 1987A-likeness is more evident in the red bands than in the blue ones. As mentioned in Sect. [discussion], data for SN 2004em have not been published yet.\n[FigA2]\n---\n1. email: andrea.pastorello@oapd.inaf.it[↩](#fnref1)\n2. *http://leda.univ-lyon1.fr/*[↩](#fnref2)\n3. *http://nedwww.ipac.caltech.edu/index.html*[↩](#fnref3)\n4. Note, however, that casted doubt on the robustness of the correlation between EW (Na ID) and interstellar extinction as determined from low resolution type Ia SN spectra.[↩](#fnref4)\n5. Images have been downloaded through the SMOKA Data Archive.[↩](#fnref5)\n6. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.[↩](#fnref6)\n7. Note that favoured time-dependent ionization effects to explain the presence of prominent Ba II lines in photospheric spectra of SN 1987A.[↩](#fnref7)\n8. We note that the luminosity of the [O I] doublet in SN 1987A and type IIP SNe remains almost constant over a long period during the nebular phase.[↩](#fnref8)\n9. Another type II SN, 2004ek, may show some similarity with this SN sub-group. Its light curve, published by, is rather peculiar, showing a sharp early-time peak, mostly visible in the bluer bands, followed by a pseudo-plateau lasting a couple of months. However, especially in the redder bands, a sort of delayed broad peak is visible, analogous to the one characterizing SNe 1982F and 2000cb. As major differences, SN 2004ek is significantly brighter (*M**V* ≈ -18) and its spectrum, showing an H*α* mostly in emission is rather peculiar. For this reason SN 2004ek has been rejected from our sample.[↩](#fnref9)\n10. The lack of photometric points during the nebular phase did not allow us to obtain reliable estimates for the 56Ni masses of SN 1982F.[↩](#fnref10)\n11. Note, however, that the distance moduli adopted by for the host of SN 2009E (*μ* = 32.41) and by and for the host of SN 2000cb (*μ* = 32.39) are different from those adopted here and reported in Table [Tab4].[↩](#fnref11)\n12. For example, the ESO/TNG large program on Supernova Variety and Nucleosynthesis Yields (*http://graspa.oapd.inaf.it/*), the Caltech Core-Collapse Program (*http://www.astro.caltech.edu/ ∼avishay/cccp.html*) or the Carnegie Supernova Project (*http://csp1.lco.cl/ ∼ cspuser1/PUB/CSP.html*)[↩](#fnref12)\n13. available via ftp through the OGLE-III (2003 season) web site *http://ogle.astrouw.edu.pl/ogle3/ews/NOOS/2003/noos.html*[↩](#fnref13)\n\nSN 2009E: a faint clone of SN 1987A\n===================================\n### Received XXXXX ; accepted XXXXX\nIn this paper we investigate the properties of SN 2009E, which exploded in a relatively nearby spiral galaxy (NGC 4141) and that is probably the faintest 1987A-like supernova discovered so far. We also attempt to characterize this subgroup of core-collapse supernovae with the help of the literature and present new data for a few additional objects. The lack of early-time observations from professional telescopes is compensated by frequent follow-up observations performed by a number of amateur astronomers. This allows us to reconstruct a well-sampled light curve for SN 2009E. Spectroscopic observations which started about 2 months after the supernova explosion, highlight significant differences between SN 2009E and the prototypical SN 1987A. Modelling the data of SN 2009E allows us to constrain the explosion parameters and the properties of the progenitor star, and compare the inferred estimates with those available for the similar SNe 1987A and 1998A. The light curve of SN 2009E is less luminous than that of SN 1987A and the other members of this class, and the maximum light curve peak is reached at a slightly later epoch than in SN 1987A. Late-time photometric observations suggest that SN 2009E ejected about 0.04 M⊙ of 56Ni, which is the smallest 56Ni mass in our sample of 1987A-like events. Modelling the observations with a radiation hydrodynamics code, we infer for SN 2009E a kinetic plus thermal energy of about 0.6 foe, an initial radius of ∼ 7 × 1012 cm and an ejected mass of ∼ 19 M⊙. The photospheric spectra show a number of narrow (v ≈ 1800 km s− 1) metal lines, with unusually strong Ba II lines. The nebular spectrum displays narrow emission lines of H, Na I, [Ca II] and [O I], with the [O I] feature being relatively strong compared to the [Ca II] doublet. The overall spectroscopic evolution is reminiscent of that of the faint 56Ni-poor type II-plateau supernovae. This suggests that SN 2009E belongs to the low-luminosity, low 56Ni mass, low-energy tail in the distribution of the 1987A-like objects in the same manner as SN 1997D and similar events represent the faint tail in the distribution of physical properties for normal type II-plateau supernovae. \nIntroduction\n============\nThe explosion of supernova (SN) 1987A in the Large Magellanic Cloud (LMC) was an epic event not only because it was the nearest SN in a period of about 4 centuries, but also because it changed significantly the general understanding of the destiny of massive stars. It was commonly believed that hydrogen-rich (H-rich) type II plateau supernovae (SNe IIP) were generated by the explosion of red supergiant (RSG) stars. However, the unusual photometric evolution of SN 1987A, with a broad light curve peak reached about 3 months after the explosion instead of the classical plateau observed in H-rich SNe, and the direct detection of the progenitor star in pre-explosion images changed this general view and proved that the precursor of this SN in the LMC was instead a blue supergiant (BSG, Arnett et al. 1989 and references therein).\nDue to their intrinsic rarity and faint early time luminosity, only very few 1987A-like objects were discovered in the past. Good datasets therefore exist only for a handful of objects (SN 1998A, Woodings et al. 1998, Pastorello et al. 2005; SN 2000cb, Hamuy 2001, Kleiser et al. 2011; SNe 2006V and 2006au, Taddia et al. 2011). In analogy with SN 1987A, some of these objects show broad, delayed light curve peaks in all bands (e.g. SN 1998A), while others (e.g. SN 2000cb and SN 1982F) show 1987A-likeness only in the red bands, while the blue bands display a rather normal type IIP behaviour.\nIn this context, the discovery of a new sub-luminous SN with a light curve comparable to that of SN 1987A, but with other observed properties (fainter intrinsic luminosity at peak, smaller synthesized 56Ni mass and lower expansion velocity of the ejected material) resembling those of some underluminous type IIP SNe (Turatto et al. 1998, Benetti et al. 2001, Pastorello et al. 2004, 2006, 2009) is even more interesting.\nThis paper is organized as follows: in Section [obs] we give basic information on SN 2009E and its host galaxy, and we present photometric and spectroscopic observations of the SN. In Section [constpro] we discuss the properties of the progenitor star and the explosion parameters as derived from the characteristics of the nebular spectrum and by modelling the SN observations. In Section [discussion] we analyse the general properties of the family of SNe similar to SN 1987A, whilst the rate of these events is computed in Section [rate]. Main conclusions are given in Section [summary]. Finally, an Appendix has been included to present our sample of 1987A-like events (Appendix [family]), and to compare the light curves of well monitored objects with those of SN 1987A (Appendix [appB]).\nObservations of SN 2009E\n========================\n [Fig1]\nThe host galaxy\n---------------\nNGC 4141 is classified by HyperLeda[2](#fn2) as an SBc galaxy, rich in H II regions (NED[3](#fn3)). estimated the integrated oxygen abundance to be log(O/H) + 12 = 8.60 (8.74 in the nucleus). They also estimated a star formation rate (SFR) from the integrated flux of H*α*, viz. ∼ 0.6 M⊙ yr− 1.\nAdopting the recessional velocity corrected for Local Group infall into Virgo quoted by HyperLeda (v*V**i**r* = 2158 ± 20 km s− 1) and a Hubble constant H0 = 72 ± 5 km s− 1 Mpc− 1, we obtain a distance of about 29.97 ± 2.10 Mpc (i.e. distance modulus *μ* = 32.38 ± 0.35 mag). The Galactic extinction in the direction of NGC 4141 is very low, i.e. *E*(*B* − *V*) = 0.02 mag. Narrow interstellar Na I *λ**λ*5889,5895 (hereafter Na ID) absorption at the host galaxy rest wavelength is marginally detected in the SN spectra, with an equivalent width (EW) of about 0.12 Å. Adopting the relation between EW*N**a**I**D* and interstellar extinction from, we find a host galaxy reddening of *E*(*B* − *V*) = 0.02 mag[4](#fn4). The total colour excess in the direction of SN 2009E is therefore *E*(*B* − *V*) = 0.04 mag.\nThe discovery of SN 2009E\n-------------------------\nSN 2009E was discovered in the spiral galaxy NGC 4141 on January 3rd, 2009, at an unfiltered magnitude of 17.8. With the distance and reddening estimated in Section [hg], the absolute magnitude at the discovery was about − 14.7. The position of SN 2009E is *α* = 12*h*09*m*49*s*.56 ± 0*s*.03, $\\delta = +58^\\circ50'50\\farcs3 \\pm 0\\farcs1$ (equinox J2000.0), which is $17\\farcs2$ East and $6\\farcs9$ South of the center of the host galaxy. According to, nothing was visible in pre-explosion images obtained on 2008 February 6th (to limiting magnitude of 19.5). We also found pre-explosion archive *g*ʹ, *R**c* and *I**c* images[5](#fn5) obtained on 2008 February 24th at the 0.5-m telescope of the Akeno Observatory/ICRR (Yamanashi, Japan) and no source brighter than *V* = 19.5, *R* = 19.6, *I* = 18.8 mag was visible at the SN position. In this paper, we will adopt January 1.0 UT (*J**D*=2454832.5, see Section [lc]) as an indicative epoch for the core-collapse.\nNGC 4141 also hosted the type II SN 2008X, that exploded $7\\farcs6$ East and $4\\farcs6$ North of the nucleus of the galaxy. Interestingly, SN 2008X was another sub-luminous event. The classification spectrum of SN 2008X obtained a few weeks after the explosion showed that it was a type II SN, although with rather narrow spectral lines, similar to those observed in SN 2005cs.\nProbably the faint apparent magnitude of SN 2009E discouraged astronomers to promptly classify this object. However, later on, noted that SN 2009E brightened by 1 − 1.5 mag during the subsequent two months, and only at about 80 days from the discovery the object was classified by as a type II SN around the end of the H recombination phase. SN 2009E appeared to show some similarity with the peculiar SN 1987A, especially in the late-time light curve brightening and the unusual strength of the Ba II P-Cygni lines. The most obvious differences were in the lower luminosity and the much narrower spectral features.\nLight Curves\n------------\nWhilst professional astronomers initially missed to follow this transient object, amateur astronomers monitored SN 2009E extensively, and it is thanks to their efforts that we can now recover information on the early flux evolution. Most images collected during the period January 2009 to May 2009 were from amateur observations. Since the end of March, we started multiband photometric and spectroscopic follow-up observations with larger, professional telescopes. All photometric data were reduced with standard techniques in IRAF[6](#fn6). Images were first overscan, bias and flat-field corrected. Then, photometric measurements of filtered images were performed using a PSF-fitting technique because suitable template images were not available. However, since the SN location was quite peripheral in the galaxy arm, its luminosity largely exceeded that of the surroundings, except at very late epochs when the SN became weak. We estimated the uncertainty due to the non-flat background by placing in all images several artificial stars close to the SN location. The adopted errors were the r.m.s. of the recovered artificial star PSF magnitudes. Multi-band zero-points for the different nights were computed through observations of standard stellar fields obtained during the same nights as the SN observations. The SN magnitudes were finally fine tuned with reference to a sequence of stars in the field of NGC 4141 (see Figure 1), calibrated by averaging magnitudes measured in selected photometric nights. The magnitudes of the local sequence stars are shown in Table [Tab1], while the final SN magnitudes are reported in Table [Tab2].\n [Fig2]\n [Fig3]\n[Tab1]\n| Star | *U* | *B* | *V* | *R* | *I* |\n| --- | --- | --- | --- | --- | --- |\n| 1 | 18.86 (0.03) | 17.59 (0.02) | 16.52 (0.02) | 15.74 (0.01) | 15.18 (0.01) |\n| 2 | 17.47 (0.02) | 16.88 (0.01) | 16.11 (0.01) | 15.60 (0.01) | 15.20 (0.01) |\n| 3 | 15.85 (0.04) | 15.87 (0.01) | 15.35 (0.01) | 15.01 (0.01) | 14.69 (0.01) |\n| 4 | 18.07 (0.03) | 16.89 (0.01) | 15.41 (0.02) | – | – |\n| 5 | 18.81 (0.03) | 18.73 (0.01) | 18.05 (0.02) | 17.62 (0.01) | 17.19 (0.02) |\n| 6 | 17.75 (0.03) | 17.85 (0.01) | 17.26 (0.01) | 16.83 (0.01) | 16.43 (0.01) |\n| 7 | 20.76 (0.05) | 19.33 (0.02) | 18.17 (0.01) | 17.31 (0.01) | 16.56 (0.02) |\n| 8 | 18.97 (0.05) | 18.91 (0.02) | 18.23 (0.01) | 17.78 (0.01) | 17.37 (0.01) |\n| 9 | – | 19.90 (0.02) | 18.46 (0.02) | 17.47 (0.01) | 16.58 (0.02) |\n| 10 | – | 16.32 (0.01) | 15.70 (0.02) | 15.31 (0.02) | 14.94 (0.01) |\n| 11 | – | 14.45 (0.02) | 13.73 (0.03) | 13.15 (0.04) | 12.83 (0.01) |\n| 12 | – | 16.43 (0.03) | 15.86 (0.03) | 15.48 (0.01) | 15.11 (0.02) |\n| 13 | 16.63 (0.06) | 16.40 (0.01) | 15.67 (0.02) | 15.22 (0.01) | 14.78 (0.02) |\n| 14 | – | 19.86 (0.06) | 18.27 (0.10) | – | – |\n [Fig4]\nSince a number of unfiltered pre-explosion images of NGC 4141 were available, magnitude measurements on the amateurs’ images were performed with the PSF-fitting technique after subtraction of the best seeing host galaxy template image. In this way we minimized the contamination of the background sources near the SN location. The subsequent photometric calibration was performed with the prescriptions discussed in. Since the detectors used by amateur astronomers in their follow-up campaign of SN 2009E have quantum efficiencies preferentially peaking at ∼ 5700 − 6500 Å, unfiltered magnitudes were scaled to match the *R*-band photometry. The SN magnitude errors accounted for the uncertainties in this convertion. The final *R*-band-calibrated magnitudes are reported in Table [Tab2], marked with an asterisk. The *B*, *V*, *R*, *I* light curves of SN 2009E, compared with those of SN 1987A, are shown in Figure [Fig2].\nIn Figure [Fig3] we compare the *B* − *V* and *V* − *I* colour curves of SN 2009E with those of a subsample of 1987A-like events and with the sub-luminous type IIP SN 2005cs. The colour curves of SN 2009E are remarkably similar to those of SN 1987A at all phases. The colour estimates at phase ∼ 25 days (*B* − *V* ≈ 1.2 and *V* − *I* ≈ 0.7 mag) suggest that, in analogy with SN 1987A, SN 2009E became red in a time scale that is shorter than for classical SNe IIP (about 1 month for SN 2009E, while ∼ 2 months for type IIP SNe). Between day 30 and day 110, during the recombination phase, the colours remain almost constant. Then, with the post-peak luminosity decline, the SN becomes redder again, reaching *B* − *V* ≈ 1.4 between 110 and 140 days after explosion (this is remarkably similar to the4 *B* − *V* colour observed in SN 1987A). The *V* − *I* colour, instead, peaks at about 200 days (*V* − *I* ≈ 1.6 mag), which is a bit earlier than observed for SN 1987A (*V* − *I* ≈ 1.4 mag between 230 and 310 days). Subsequently, the colours turn back to the blue again.\nThe *R*-band absolute light curve of SN 2009E is shown in Figure [Fig4], together with those of a number of other type II SNe with a broad, delayed light curve peak, similar to that observed for SN 1987A. In analogy with the early discovery of SN 1987A, also SN 2009E was likely caught soon after the core-collapse. Marginal evidence of the initial sharp peak due to the shock breakout is possibly visible (Figure [Fig4], panel C), although the large error bars of the unfiltered amateur images make this finding rather uncertain.\nNevertheless, we can constrain the explosion epoch of SN 2009E with a small uncertainty through a comparison with the early photometric evolution of SN 1987A. Hereafter we will adopt *J**D*=2454832.5− 5+ 2 as the best estimate for the core-collapse epoch of SN 2009E. The distances and reddenings used for our sample of 1987A-like events shown in Figure [Fig4] are listed in Tables [Tab4] and [Tab5]. SN 2009E is the faintest SN in this sample at all the epochs. In particular, its peak magnitude *M**R* ∼ − 16.2 is about 0.4 mag fainter than that of SN 1987A and its 56Ni mass, as derived from the luminosity of the radioactive tail relative to that of SN 1987A, is 0.040− 0.011+ 0.015 M⊙, the lowest among the objects in our sample.\nAnother remarkable property of SN 2009E is that its *R*-band light curve peaks at about +96d after core-collapse, significantly later than SN 1987A ( ∼ +84 days). This suggests different ejecta parameters for SN 2009E (e.g. smaller initial radius, lower expansion velocity or a more massive envelope, see Section [model]). Interestingly, fainter SNe in our sample seem to reach peak magnitude later than the more luminous ones (see also Appendix [appB]).\nSpectra\n-------\n[Tab3]\n| Date | Average *J**D* | Phase⋆ | Instrumental configuration | Exposure (s) | Resolution (Å)$^\\medbullet$ | Range (Å) |\n| --- | --- | --- | --- | --- | --- | --- |\n| Mar24, 2009 | 2454915.40 | +82.9 | Ekar1.82m+AFOSC+gr.4 | 3600 | 24 | 3490–7790 |\n| Mar25, 2009‡ | 2454915.54 | +83.0 | Asiago1.22m+B&C+g600 | 3600 | 6.1 | 5090–7500 |\n| Apr01, 2009 | 2454922.52 | +90.0 | NOT+ALFOSC+gr4 | 1200 | 13 | 3360-9100 |\n| Apr12, 2009 | 2454934.48 | +102.0 | NOT+ALFOSC+gr4 | 1200 | 16 | 3230-9060 |\n| Apr16, 2009 | 2454938.42 | +105.9 | CAHA2.2m+CAFOS+g200 | 1792 | 9.5 | 3700-10670 |\n| May04, 2009 | 2454955.57 | +123.1 | NOT+ALFOSC+gr4 | 2700 | 13 | 3350-9120 |\n| May13, 2009 | 2454964.62 | +132.1 | TNG+LRS+LRR | 3600 | 12 | 5050-9520 |\n| May20, 2009 | 2454971.55 | +139.1 | TNG+LRS+LRR | 3600 | 12 | 4960-9560 |\n| Jun26, 2009‡ | 2455009.42 | +176.9 | CAHA2.2m+CAFOS+g200 | 2 × 2400 | 9.5 | 4100-9100 |\n| Jul18, 2009 | 2455031.44 | +198.9 | CAHA2.2m+CAFOS+g200+GG495 | 2 × 3600 | 13 | 4800-9690 |\n| Aug15, 2009† | 2455058.84 | +226.3 | CAHA2.2m+CAFOS+g200 | 2 × 3600 | 12 | 3880-10670 |\n| Jan22, 2010 | 2455218.64 | +386.1 | NOT+ALFOSC+gr5 | 3 × 1200 | 14 | 5000–10000 |\n| Jan24, 2010 | 2455220.70 | +388.2 | NOT+ALFOSC+gr4 | 3 × 1200 | 12 | 3550–9080 |\n⋆ Days from explosion (*J**D*=2454832.5); $^\\medbullet$ As estimated from the full width at half maximum of isolated night sky lines. ‡ Poor signal-to-noise spectrum; † Weighted average of 2 spectra obtained on August 14th and 15th.\n [Fig5]\nAs mentioned above, the spectroscopic classification of SN 2009E was performed about 80 days after the SN discovery. To our knowledge, no spectra exist to witness the early time evolution of this object. After the SN classification, we started a regular spectroscopic monitoring covering the SN evolution from near maximum light to the nebular phase. Spectra have been reduced in the IRAF environment using standard prescriptions. After the optimal extraction, the SN spectra were wavelength calibrated using arc lamp spectra and then flux calibrated using flux standard stars observed in the same nights as the SN. A final check with photometry was performed in order to fine tune the flux calibration of the spectra, with a remaining uncertainty in the flux calibration of less than 10%. Basic information on the spectra collected for SN 2009E is listed in Table [Tab3], while the spectral sequence until August 15, 2009 is shown in Figure [Fig5].\nSince the earliest spectrum was obtained about 83 days after the core-collapse, the continuum is quite red, indicative of relatively low temperatures (T*b**b* ≈ 6000 K). All the photospheric spectra show a clear flux deficit at the blue wavelengths (below 4500 Å) that is due to line blanketing from increasingly strong metal lines. These spectra are rich in narrow P-Cygni lines indicating remarkably low velocity of the expanding ejecta. From the position of the minima of the Sc II *λ*6245 and *λ*5527 lines, we measured v*p**h* ≈ 1600 km s− 1 in our earliest spectrum and about 1000 km s− 1 at the end of the photospheric phase (Figure [Fig6]). In the same period, the velocity of the unblended Ba II *λ*6142 line declines from 2200 km s− 1 to 1850 km s− 1.\n [Fig6]\nIn Figure [Fig7], a late time photospheric spectrum of SN 2009E (phase about 3 months after explosion) is compared with spectra of SN 1987A and the 1987A-like event SN 1998A, and with spectra of faint SNe IIP. While the strong Ba II lines are a common feature both for the spectra of SN 1987A and under-luminous, 56Ni-poor type IIP SNe, the narrowness of the spectral lines in SN 2009E is more reminiscent of 1997D-like events.\n [Fig7]\n [Fig8]\nTogether with quite a weak H*α*, we clearly identify prominent lines of Na ID, Ca II, O I, Fe II, Ti II, Sc II and Cr II [see for a detailed line identification for underluminous SNe IIP during the photospheric phase, and for the nebular phase]. The Ba II lines (multiplet 2) are among the strongest spectral features. This is a common characteristic of both SN 1987A and faint SNe IIP. The presence of prominent Ba II lines in the photospheric spectra of SN 1987A was interpreted by as a signature of an over-abundance of Ba in the outer layers of the SN [7](#fn7). However, the presence of prominent Ba II features in underluminous SNe IIP can also be explained invoking temperature effects. The evolution of Na ID is puzzling. The absorption component becomes broader with time (see Figure [Fig5] for the evolution of this spectral feature from 2009 April 16 to May 13). A similar behaviour, although less extreme, is observed also for H*α*. This effect is probably due to line blending, with the Ba II *λ*5854 and *λ*6497 lines (and possibly of other metal features) having an increasing strength relative to Na ID and H*α* respectively. The evolution of the expansion velocities, as deduced from the minima of the P-Cygni of H*α*, Na I, Ba II, Fe II and Sc II, is shown in Figure [Fig6]. Metal lines show a monotonic decline and velocities at the end of the plateau phase (1200 − 1800 km s− 1) that are only marginally higher than those observed in sub-luminous type IIP SNe, whilst H*α* and Na ID display an opposite trend. Since May (around 4 months past explosion, Figure [Fig5]), the classical nebular doublets of [Ca II] *λ**λ* 7291-7323 and [O I] *λ**λ* 6300-6364 become visible, and increase in strength with time.\nA late time spectrum of SN 2009E was obtained at the Nordic Optical Telescope (NOT) about 1 year after explosion (see Figure [Fig8]). Although the transition toward a purely nebular appearence is not complete (e.g. broad P-Cygni features of Na ID, O I *λ*7774 and Ca II NIR are still visible), the spectrum is clearly dominated by prominent H*α* in emission, [O I] *λ**λ*6300,6364 and [Ca II] *λ**λ*7291,7323. In addition, a number of [Fe II] features are detected. In this phase the spectrum of SN 2009E more closely resembles those of SN 1987A and SN 1998A (Figure [Fig8], top), although the spectral lines are significantly narrower than those of other 1987A-like events and are closer to those of sub-luminous, 56Ni-poor SNe (Figure [Fig8], bottom).\nConstraining the progenitor of SN 2009E\n=======================================\nThe ejected oxygen mass\n-----------------------\nThe strength of the [O I] *λ**λ*6300,6364 lines in the nebular spectra of core-collapse SNe can be used to roughly estimate the O mass ejected in the SN explosion. This method has been applied for determining the O masses for a few type IIP SNe. We first make an attempt to estimate the minimum O mass needed to produce the [O I] *λ**λ* 6300-6364 feature with the relation between the O mass and the total flux of the doublet presented by. Using the nebular spectrum of SN 2009E, we measure an observed flux F[*O**I*] ≈ 3.35 × 10− 15 erg s− 1 cm− 2. Accounting for the reddening and distance estimates as mentioned in Section [hg], we obtain for SN 2009E a minimum O mass in the range between 0.05 M⊙ and 0.22 M⊙. As a comparison, derived a minimum O mass range for SN 1987A of 0.3 − 1.1 M⊙. The only other 1987A-like object with nebular spectra available is SN 1998A. Using the spectrum at phase ∼ 344 days (see Figure [Fig8], top), we obtain an O mass in the range 0.24 − 1.01 M⊙.\nIn order to improve the O mass estimate, we adopt the method already used in, which is based on the fact that the luminosity of the [O I] doublet in the nebular phase is known to be powered by the *γ*-rays produced in the radioactive decays of 56Co to 56Fe, and deposited in the O-rich ejecta. Since the mass of O in SN 1987A is reasonably well-known, we can derive the O masses of SNe 2009E and 1998A via comparison with the O mass of SN 1987A. We use the relation linking the O mass with the luminosities of the [O I] doublet[8](#fn8) and of the radioactively powered light curve tail, computed at similar phases :\n \n$$\\label{eq1}\nL\\_{[OI]} = \\eta\\_O \\frac{M\\_O}{M\\_{exc}} L\\_{^{56}Co},$$ \nwhere L[*O**I*] and L56*C**o* are the luminosities of the [O I] doublet and of the radioactive tail (respectively), M*O* is the mass of O, M*e**x**c* represents the whole excited mass in which the radioactive energy is deposited, and *η**O* is the efficiency of the transformation of the deposited energy into [O I] line radiation. Assuming that M*e**x**c* and *η*0 are similar in all 1987A-like events and noting that (since the late light curve of SN 2009E follows the 56Co decay rate)\n \n$$\\frac{L\\_{^{56}Co}(1987A)}{L\\_{^{56}Co}(SN)} \\propto \\frac{M\\_{^{56}Ni}(1987A)}{M\\_{^{56}Ni}(SN)},$$ \nfrom Equation [eq1], we obtain:\n \n$$\\label{eq3} \nM\\_O(SN) = M\\_O(1987A) \\times \\frac{L\\_{[OI]}(SN) \\times M\\_{^{56}Ni}(1987A)}{L\\_{[OI]}(1987A) \\times M\\_{^{56}Ni}(SN)}.$$ \nNow we have all the ingredients needed to estimate the O mass in SNe 2009E and 1998A from Equation [eq3] using the information available for SN 1987A. For SN 2009E we obtain M*O* ≈ 0.60 − 0.75 M⊙ (using as O mass for SN 1987A the extreme values of M*O*, *m**i**n* = 1.2 M⊙ and M*O*, *m**a**x* = 1.5 M⊙, respectively), while for SN 1998A we obtain M*O* ≈ 1.18 − 1.48 M⊙. Therefore, with this method, we determine for the under-luminous SN 2009E an ejected mass of O that is a factor of 2 smaller than that of SN 1987A (note that for SN 2009E also the 56Ni mass is a factor of ∼ 2 smaller). In the case of SN 1998A we obtain an O mass that matches the values of SN 1987A, despite the former object ejected more 56Ni.\nAccording to the models presented by and, a star with main sequence mass of about 18 − 20 M⊙ produces 1.2 − 1.5 M⊙ of oxygen and a 56Ni mass of 0.07-0.1 M⊙. This is in excellent agreement with the values observed in SNe 1987A and 1998A. It is more difficult to provide an interpretation for the abundances displayed by SN 2009E (both M*O* and M56*N**i* are smaller by a factor 2 than those of SN 1987A), although we may tentatively guess that the lower O and 56Ni masses would point toward a lower-mass main sequence star. We remark that the above estimates have been obtained under the simplifying assumption that M*e**x**c* and *η**O* were the same for SNe 1987A, 1998A and 2009E. This is not necessarily true, since we can reasonably expect a range of values for these parameters in 1987A-like events, and this would significantly affect our O mass estimates.\nModelling the data of SN 2009E\n------------------------------\nOther physical properties of SN 2009E (namely, the ejected mass, the progenitor initial radius and the explosion energy) have been estimated by performing a model vs. data comparison using two codes that compute the bolometric light curve, the evolution of the ejecta velocity and the continuum temperature at the photosphere of a SN. The first code, based on a simplified semi-analytic treatment of the ejecta evolution, is used to perform a preparatory study in order to constrain the parameter space. The second code, that includes a more accurate treatment of radiative transfer and radiation hydrodynamics, computes the tighter grid of models needed for the final comparison. This is similar to the procedure adopted for SN 2007od in.\n [Fig9]\nThe bolometric light curve was computed from the observed multicolor light curves using the assumptions on extinction and distance modulus reported in Section [hg]. We have adopted the explosion epoch estimated in Section [lc] (*J**D* = 2454832.5− 5+ 2). Finally, we have assumed that SN 2009E had the same color evolution as SN 1987A (this is a reasonable assumption, see e.g. Figure [Fig3]), and hence\n \n$$L(2009E) = L(1987A) \\times \\frac{L\\_{BVRI}(2009E)}{L\\_{BVRI}(1987A)},$$ \nwhere *L* and *L**B**V**R**I* are the bolometric and quasi-bolometric (*B**V**R**I*) luminosities of the two SNe.\nThe best *χ*2 fit to the bolometric light curve of SN 2009E obtained with the full radiation-hydrodynamics code returned values of a total (kinetic plus thermal) energy (E) of 0.6 foe, an initial radius (R0) of 7 × 1012 cm, and an envelope mass (M*e**j*) of 19*M*⊙ (see Figure [Fig9]). As a comparison, the fit with the semi-analytic code gave E = 1.3 foe, R0 = 6 × 1012 cm, and M*e**j* = 26 M⊙ (see again Figure [Fig9]). The parameters inferred from the two best-fits are in reasonable agreement. The envelope mass (and consequently the total energy) are slightly overestimated by the semi-analytic code as a consequence of the different mass distribution. Indeed, given the assumption of uniform density throughout the ejecta, the semi-analytic code tends to gather more mass (and hence more energy) in the external, faster-moving layers. The 56Ni distribution also affects the mass estimate because, as a consequence of the simplified uniform distribution adopted in the semi-analytic code, less energy is released at the end of the photospheric phase and, hence, a more massive envelope is needed to sustain the recombination phase.\nThe ejected 56Ni mass of 0.039 M⊙ that is adopted in the semi-analytic code is in an excellent agreement with that estimated in Section [lc], whilst the one inferred from the radiation-hydrodynamic code is slightly higher (0.043 M⊙). Since the latter code accounts also for the amount of material which (eventually) falls back onto the central remnant during the post-explosive evolution, the numerical simulations need a larger initial 56Ni mass to reproduce the observed late-time light curve of SN 2009E.\nIn Figure [Fig9] the evolutions of the photospheric velocity and temperature are also shown. The agreement between models and observations in the late photospheric phase is good, with possibly some discrepancy in matching the photospheric velocities at later times ($\\gtrsim$ 4 months). The decline of the model velocity profile is faster than that inferred from the observations, probably because some extra radioactive decay heating occurs in the inner part of the ejecta of SN 2009E as a consequence of a different (i.e. more centrally condensed) distribution of 56Ni.\n[Tab4]\n| SN | Host galaxy | Type1 | *μ*2 | A*B*, *M**W* | *M**B* | d*S**N*0/r25 | P04 | P04 corr. |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| 1909A | NGC 5457 | SABc | 29.343 | 0.037 | -21.01 | 1.119 | 8.59 | 8.3010 |\n| 1982F | NGC 4490 | SBcd | 29.92 | 0.093 | -20.25 | 0.23 | 8.53 | 8.6011 |\n| 1987A | LMC | SBm | 18.504 | 0.324 | -18.02 | - | 8.31 | - |\n| 1998A | IC 2627 | SABc | 32.19 | 0.518 | -20.03 | 0.66 | 8.51 | 8.41 |\n| 1998bt | A132541-2646 | - | 36.415 | 0.244 | -13.25 | - | 7.98 | - |\n| 2000cb | IC 1158 | SABc | 32.686 | 0.491 | -19.34 | 0.93 | 8.46 | 8.24 |\n| NOOS-005 | 2MASX J05553978-6855381 | S | 35.46 | 0.6568 | -20.01 | 1.26 | 8.51 | 8.17 |\n| 2004em | IC 1303 | Sc | 34.07 | 0.466 | -19.73 | 0.96 | 8.49 | 8.26 |\n| 2005ci | NGC 5682 | Sb | 32.73 | 0.141 | -18.07 | 0.66 | 8.36 | 8.26 |\n| 2006V | UGC 6510 | SABc | 34.367 | 0.125 | -20.97 | 1.31 | 8.59 | 8.22 |\n| 2006au | UGC 11057 | Sc | 33.387 | 0.742 | -20.64 | 0.90 | 8.56 | 8.36 |\n| 2009E | NGC 4141 | SBc | 32.38 | 0.086 | -17.75 | 0.68 | 8.33 | 8.22 |\n1 Morphologic type as quoted by HyperLeda. \n2 Unless otherwise specified, distance moduli are computed from HyperLeda’s v*V**i**r*, adopting H0 = 72 km s− 1 Mpc− 1. \n3 Distance computed from the tip of the red giant branch. Cepheid distances are found to be too sensitive to metallicity. The metallicity correction is even more crucial in the case of NGC 5457 where inner metal-poor and outer metal-rich cepheids give uncorrected distance moduli that may differ by 0.3-0.4 mags. \n4 For LMC we adopted a cepheid distance, well consistent with that derived from the TRGB. \n5 SN 1998bt was discovered in the course of the Mount Stromlo Abell cluster supernova search. Unfortunately, no spectroscopic classification exists for this SN. \nAccording to, there is a very faint host galaxy at the position of SN 1998bt, but no spectrum was ever obtained of this galaxy. We assume that the galaxy hosting SN 1998bt belongs to a cluster monitored by the search whose average redshift is z = 0.046. \n6 Average EPM distance from Table 4.1 of. \n7 Distance as in, but obtained adopting an Hubble Constant H0 = 72 km s− 1 Mpc− 1. \n8 This galaxy is behind LMC, so the extinction reported here includes both Galaxy and LMC contributions (see text for references). \n9 Computed using the position angle provided by. \n10 Other sources give 12+log(O/H) ≈ 7.7-7.8 at the position of SN 1909A. \n11 An alternative estimate from gives 12+log(O/H) = 8.35. \n[Tab5]\nccccccccccc SN & E*t**o**t*(B-V)*a* & *J**D**e**x**p**l*. & Band & *J**D**p**e**a**k* & t-t0 (days) & *m**p**e**a**k*(filter) & *M**p**e**a**k* & M(56Ni) & Sources \n1909A & 0.009 & 2418310 ± 30 & *B* & 2418383 ± 7 & 73 ± 31 & 14.47 ± 0.30 & -15.91 & 0.13M⊙*b* & 1 \n1982F & 0.022 & 2445014 ± 15 & *B* & 2445087 ± 4 & 73 ± 16 & 16.40 ± 0.02 & -13.61 & – & 2,3,4 \n& & & *V* & 2445097 ± 3 & 83 ± 15 & 14.95 ± 0.10 & -15.04 & & \n1987A & 0.19 & 2446849.816*c* & *B* & 2446932.0 ± 1.1 & 82.2 ± 1.1 & 4.52 ± 0.01 & -14.77 & 0.075M⊙ & 5 \n& & & *V* & 2446932.2 ± 1.0 & 82.4 ± 1.0 & 2.96 ± 0.01 & -16.13 & & \n& & & *R* & 2446933.6 ± 1.0 & 83.8 ± 1.0 & 2.27 ± 0.01 & -16.67 & & \n& & & *I* & 2446935.2 ± 0.9 & 85.4 ± 0.9 & 1.90 ± 0.01 & -16.88 & & \n1998A & 0.120 & 2450801 ± 4 & *B* & 2450876.3 ± 7.4 & 75.3 ± 8.4 & 17.59 ± 0.13 & -15.12 & 0.09M⊙ & 6 \n& & & *V* & 2450885.4 ± 8.3 & 84.4 ± 9.2 & 16.31 ± 0.09 & -16.28 & & \n& & & *R* & 2450885.6 ± 3.9 & 84.6 ± 5.6 & 15.73 ± 0.06 & -16.78 & & \n& & & *I* & 2450889.2 ± 5.5 & 88.2 ± 6.8 & 15.37 ± 0.07 & -17.05 & & \n1998bt & 0.057 & 2450831 ± 30 & *V* & 2450905 ± 6 & 74 ± 31 & 19.81 ± 0.05 & -16.79 & & 7,0 \n& & & *R* & 2450908 ± 8 & 77 ± 31 & 19.38 ± 0.08 & -17.18 & & 8,0 \n2000cb& 0.112 & 2451653.8 ± 1.4 & *B**d* & & & 18.23 ± 0.02 & -14.94 & 0.11M⊙ & 8,9,10 \n& & & *V* & 2451714.6 ± 4.7 & 60.8 ± 4.9 & 16.58 ± 0.02 & -16.47 & & \n& & & *R* & 2451725.2 ± 2.1 & 71.4 ± 2.5 & 15.95 ± 0.02 & -17.03 & & \n& & & *I* & 2451729.0 ± 1.4 & 75.2 ± 2.0 & 15.56 ± 0.02 & -17.34 & & \nNOOS-005&0.160 & 2452855 ± 15 & *I* & 2452939.5 ± 2.7 & 84.5 ± 15.2 & 18.22 ± 0.02 & -17.51 & 0.13M⊙ & 0 \n2005ci & 0.033? & 2453521 ± 3 & *u**n**f*. & & & < 17.5 & < -15.3 & & 10 \n2006V*e* & 0.029 & 2453748 ± 4 & *B* & 2453823.7 & 75.7 & 18.29 ± 0.01 & -16.20 & 0.13M⊙ & 11 \n2006au*e* & 0.312 & 2453794 ± 9 & *B* & 2453865.5 & 71.5 & 18.63 ± 0.02 & -16.06 & < 0.07M⊙ & 11 \n2009E & 0.04 & 2454832.5 ± 2.0 & *B* & 2454919.0 ± 6.5 & 86.5 ± 6.8 & 18.01 ± 0.03 & -14.54 & 0.04M⊙ & 0 \n& & & *V* & 2454924.9 ± 5.9 & 92.4 ± 6.2 & 16.76 ± 0.04 & -15.74 & & \n& & & *R* & 2454928.3 ± 2.8 & 95.8 ± 3.4 & 16.25 ± 0.02 & -16.21 & & \n& & & *I* & 2454928.3 ± 6.0 & 95.8 ± 6.3 & 15.98 ± 0.03 & -16.46 & & \n*a* The SN extinction is computed accounting for both Milky Way and host galaxy reddening components; in the case of NOOS-005, since the host galaxy is projected behind LMC, we also accounted for the contribution of LMC. \n*b* Computed using the B-band detections at days +429 and +432, and assuming the same colour evolution as SN 1987A. Using the detection at day +216, we would obtain M(56Ni) ≈ 0.35M⊙. \n*c* and references therein. \n*d* The broad delayed maximum typical of 1987A-like SNe is not visible in the B band; here we report the magnitude of the pseudo-plateau visible after the early time B-band maximum. \n*e* For a comprehensive information on the multi-band light curve parameters, see Table 7 of. \n0 = This paper; 1 = Sandage & Tammann 1974; 2 = Yamagata & Iye 1982; 3 = Tsvetkov 1984; 4 = Tsvetkov 1988; 5 = Whitelock et al. 1989 (and references therein); 6 = Pastorello et al. 2005; 7 = Germany et al. 2004; 8 = Hamuy 2001; 9 = Hamuy & Pinto 2002; 10 = Kleiser et al. 2011; 11 = Taddia et al. 2011.\nFinally, adopting a mass of the compact remnant of about 2 M⊙ (and assuming a negligible pre-SN mass loss), we derive a final mass of 21 M⊙ for the progenitor of SN 2009E, which is slightly higher than our rough estimate obtained throught the O mass estimate (Section [OI]).\nSN 2009E and other 1987A-like events\n====================================\nAlthough 1987A-likeness was claimed for a number of objects in the past, reasonable observational evidence exists for only a small number of events [9](#fn"},"data_id":{"kind":"string","value":"arxiv_0182794"}}},{"rowIdx":45699,"cells":{"content":{"kind":"string","value":"ability_concreteness_flip.png)*stability* measure vs. *concreteness* scores.\nAnalysis\n========\nConcept stability\n-----------------\nFigure [fig:stabilityconcreteness] and Table [tab:concretenessstabilitybinary] support our hypothesis that concreteness contributes to stability, as most concrete concepts tend to be stable and most abstract concepts tend to be less stable. However, recall is higher than precision in Table [tab:concretenessstabilitybinary], indicating that stable concepts are mostly concrete but not vice versa, as plenty of concepts are located in the lower right in Figure [fig:stabilityconcreteness]. In our analysis, we find that some concrete concepts are unstable because meaning is extended by semantic processes like metonymy and metaphor. We now give examples of our analysis of this phenomenon in. See for the visual semantic fields of all 83 concepts.\nshows (i) concrete concepts whose instability is plausibly caused by metaphor, metonymy, and other semantic extensions and (ii) crosslingually associated English concepts that indicate instability. For example, metonymy, where a concept is represented by the name of something associated with it, results in instability frequently. This is in line with crosslinguistic studies that show that metonymy seems to be a universal language phenomenon.\nlll types & concept & associated \n&‘ear’ & *hear*, *listen* \n& ‘mouth’ & *word*, *speak* \n & ‘sun’ & *day*, *east* \n& ‘moon’ & *month* \n& ‘belly’ & *womb*, *birth* \n& ‘soldier’ & *army*, *military* \n & ‘seed’ & *son*, *offspring* \n& ‘mouth’ & *entrance* \n & ‘water’ & *waters*, *sea*, *drink* \n& ‘knee’ & *obeisance*, *worship* \nProcesses of semantic extension are central to cognitive linguistics for explaining historical changes, language use and framing in discourse. Since these processes are pervasive in communication, in the lexicon and in historical language change, they are also a plausible cause for crosslingual instability of concepts.\nc\n[fig:clustering]\nLanguage similarity\n-------------------\nshows a visualization of conceptual similarity using t-SNE. Each point in the figure represents a language. Color indicates family on the left (for the six families from ) and area on the right (for all languages, area extracted from Glottolog 4.7).\nThe right figure (area) shows that conceptual similarity is correlated with geographic proximity: except for, all regions are concentrated in a few subclusters, with being most clustered and least. For example, there are two clusters (top left, top center) that cover most languages. This is consistent with findings that close languages often influence each other. Of course, the main reason is the correlation between conceptual and typological relatedness (see below) and between typological and geographic proximities. But there are also indications of geography-induced conceptual similarity, e.g., most languages have close neighbors. We present examples at the end of.\nThat geographic proximity is not sufficient for conceptual similarity is demonstrated by : it is distributed over most of the figure, indicating that the underlying conceptualizations are quite different. This is consistent with the well-known fact that exhibits enormous linguistic diversity in a very small geographic area. Of course, there is still subcluster structure: most languages have near neighbors in conceptual space.\nTurning to the left figure (family), we see that there is agreement of conceptual similarity with typological similarity. Some families form a relatively clear cluster, in particular,, suggesting that languages are similar in conceptualization. This explains ’s high accuracy in.\nBut there is also complementarity between conceptual similarity and typological similarity. is spread out across the entire figure, explaining its low accuracy in. For Chinese and Tibetan, we find their conceptualizations to be quite different, in particular, for body parts (e.g., `mouth` and `neck`). See for examples.\nConversely, we now present three examples of typologically distant languages that are still conceptually close. The first example is **Tagalog**. While languages mostly occur in a relatively tight region of the figure, that region also contains many non-Indo-European languages. We hypothesize that languages have influenced the conceptualization of other languages worldwide due to their widespread use, partly as part of colonization. One example is that the Tagalog words and mean both and, a conceptualization similar to Spanish (), a language Tagalog was in close contact with for centuries. Standard Malay is typologically related to Tagalog, but its word for, does not include the meaning. This may contribute to Tagalog being conceptually more similar to Spanish on our measure than other languages. **Plateau Malagasy**, an language spoken in Madagascar, is conceptually similar to both far-away languages like Hawaiian (reflecting its typology) as well as to geographically close, but typologically dissimilar languages like Mwani and Koti. **Masana** is an Afro-Asiatic language spoken in Nigeria. It is conceptually close to the languages Yoruba, Igbo and Twi, also spoken in and around Nigeria. Geographic proximity seems to boost conceptual similarity in these three cases. We leave further investigation of the hypothesis that -based representations reveal historical interlanguage influences to future work.\nConclusion & Future Work\n========================\nWe propose, a method that automatically aligns source-language concepts and target-language strings by creating a directed bipartite graph. We investigate the structure of such alignments for 83 focal concepts. Our extensive manual evaluation demonstrates good performance of. We introduce the notion of crosslingual stability of a concept and show, using, that concrete concepts are more stable across languages than abstract concepts. We also define conceptual similarity, a new measure of language similarity based on representations. In our experiments, conceptual similarity gives results that partially agree with established measures like typological and areal similarity, but are complementary in that they isolate a single clearly defined dimension of similarity: the degree to which the conceptualization of two languages is similar.\nIn the future, we would like to improve the efficiency of and, extending our work on a sample of 83 in this paper, apply it to all concepts that occur in PBC.\nLimitations\n===========\nThe we propose consists of two core steps, i.e., forward pass and backward pass. The forward pass identifies the most associated target-language strings for a focal concept. However, due to possible data sparsity of PBC in some low-resource languages and some cases of verse-level misalignment, *χ*2 scores of the real translations can be indistinguishable compared with some other rare words that also occur in the same verses. Under such rare cases, will not work well enough. In addition, the genre of PBC is limited to religion and therefore the diversity of the concepts across languages is largely influenced. Nevertheless, PBC, as far as we know, provides texts in the largest number of low-resource languages. PBC is thus a good fit for our goal.\nIn this work, we select 83 concepts, including the Swadesh32 and Bible51, representing a wide range of interesting crosslingual concepts. The runtime for computing the results for one concept in all languages is around 10 hours on average. The relatively long runtime, however, can prevent us from exploring more interesting concepts.\nWe find that the concreteness of a focal concept can be a contributor to the stability measure. As we use English as the source language for representing the focal concepts, we naturally resort to concreteness scores from English language ratings only. In addition, the analysis is carried out from an English perspective. Nevertheless, as we want to compare different languages, we have to use a unified source language. Theoretically, we can use any language as the source language and represent the concepts in that language. We therefore plan to use other languages, e.g., Chinese, or some low-resource languages, as the source language in future research.\nEthics Statement & Risks\n========================\nIn this work, we investigate the differences in conceptualization across 1,335 languages by aligning concepts in a parallel corpus. To this end, we propose, a method that creates a directed bipartite alignment graph between source language concepts and sets of target language strings. The corpus we used, i.e., PBC, contains translations of the Bible in different languages (one language can have multiple editions). As far as we know, the corpus does not include any information that can be used to attribute to specific individuals. Therefore, we do not foresee any risks or potential ethical problems.\nAcknowledgments\n===============\nWe would like to acknowledge Verena Blaschke for her valuable suggestions. We would also like to thank the reviewers for their positive and constructive feedback. This work was funded by the European Research Council (grant #740516).\nDetails of data\n===============\nParallel Bible Corpus\n---------------------\nWe work with the Parallel Bible Corpus (PBC, ), which contains 1,775 editions of the Bible in 1,335 unique languages (we regard dialects with their own ISO 639-3 codes[3](#fn3) as different languages). As far as we know, there is no explicit licence for PBC dataset. For each language, we only use one of its available editions. We use the *New World* edition for each language, if available, and the edition with the largest number of verses otherwise. Different from previous work which only used verses that are available in all languages, we use all parallel verses between English and any other target languages. This means the number of parallel verses between English and other languages can be different. In general, we have parallel verses with a number greater than 30,000 (Hebrew + New Testament) for high-resource languages (141 languages), e.g. French, German and Chinese, while around 7,900 (New Testament only) for most of the other languages (1038 languages).\nConcept selection\n-----------------\nWe have 83 focal concepts in total which are listed in Tables [tab:conceptstability1] and [tab:conceptstability2]. We classify the focal concepts into Swadesh32 and Bible51, for which we explain the selection of concepts in detail as follows.\nThe *Swadesh 100* list offers 100 English words that represent universal and basic concepts. The words include nouns, adjectives, and verbs. We limit our selection to nouns to facilitate the comparison between concepts. Because we choose Bible editions as our resource, many concepts in the list can have very low frequencies of occurrences or even do not occur at all. On the contrary, some concepts in the list can occur many times but are less interesting to us, e.g., ‘I’, ‘you’, ‘we’, ‘this’ etc. We therefore only keep the concepts in the list which occur equal to or more than 5 times in the New Testament (only New Testament is available for most low-resource languages so the concept has to appear in the New Testament) and less than or equal to 500 times in Hebrew + New Testaments. There are 32 concepts that fulfill the criterion and we refer them to as **Swadesh32**, which are shown in Table [tab:conceptstability1].\nFor Bible concepts, we first obtain the distinct types of strings that have a length between 4 and 15 characters from all words in the English Bible. The ngrams can contain but cannot go beyond the word boundaries. For example, from a part of sentence `$bird$fly$` (substituting whitespaces with `$`), we can obtain ngrams such as `$bird`, `$bird$` and `fly$`, but `$bird$f` is not possible because it contains `$` in the middle. After this, we randomly select 10 languages from the available languages of PBC plus Chinese and German (12 languages in total). For each of the selected languages, we compute the coverage of the identified most associated target-language string obtained by performing a forward pass (setting the max number of iterations *M* to 1) (Algorithm [alg:fs]) by regarding each English string as a focal concept. If the coverage of a string is larger than 0.5 for more than five languages, then we keep it otherwise we filter it out. This results in around 1000 strings. Then we filter out those that represent named entities. Finally, we manually check the list and select the strings that represent nouns and are not in the Swadesh list and are more or less specific to the Bible. This finally results in a set of 51 concepts (**Bible51**) which are shown in Table [tab:conceptstability2].\nAdditional details of\n=====================\nFocal concepts & strings\n------------------------\nThe bipartite graph $\\cal G$ we construct contains source nodes set $\\cal S$ and target nodes set $\\cal T$. Each node $s \\in \\cal{S}$ is a concept and is represented by a set of strings. We restrict the length of the strings between 1 and 8 for any language except the source language English (we restrict the length larger than 2 and the strings cannot go beyond word boundaries) for efficiency. To differentiate the nodes in ${\\cal S}$, we refer to the set of our chosen 83 focal concepts as ${\\cal S}\\_F \\subset {\\cal S}$. Each focal concept *F* in ${\\cal S}\\_F$ is a set which can contain multiple strings, e.g., ‘belly’ concept: {`\\$belly\\$`, `\\$bellies\\$`}. In contrast, other concepts in ${\\cal S}$ are sets which contain only a single string, e.g., {`\\$sparrows\\$`}. In the backward pass for a focal concept *F*, if a string *s* being identified belongs to *F*, then we create an edge that ends at *F* instead of *s*.\nString candidates\n-----------------\nThe consists of (1) a forward pass: for building edges $(F, c) \\in {\\cal S} \\times \\cal{T}$ and (2) a backward pass for building edges $(c, f) \\in {\\cal T} \\times {\\cal S}$. In the forward pass, for example, an edge $(F, c) \\in {\\cal S} \\times {\\cal T}$ is constructed if a target node from the target language *l*: *c* = < *l*, *v*, *T* > ’s verse *v* contains a string *t* that is highly associated with *F*. As the search space of target-language strings is extremely large for each *l*, we therefore restrict the search space to the set of strings which occur in the verses whose corresponding English verses contain *F*. Formally, let Π(eng, *F*) be the verses where the focal concept *F* occurs and ${\\cal P}(\\Sigma^\\*)(l, v)$ be the strings that fulfill our string selection conditions in verse *v* for language *l*. We will then only consider the strings in $\\mathbb{T} = \n\\bigcup\\_v {\\cal P}(\\Sigma^\\*)(l, v) | v \\in \\Pi(\\text{eng}, F)$ to be candidates in language *l* that are possibly associated with *F*. Similarly, in the backward pass, we also restrict the search space to be the set of English strings in the verses whose corresponding target-language verses contained the identified target-language string set *T* in the forward pass. Formally, let Π(*l*, *T*) be the verses where the *T* occurs and ${\\cal P}(\\Sigma^\\*)(\\text{eng}, v)$ be the strings that fulfill our string selection conditions in verse *v* for English. We will then only consider English strings in $\\mathbb{S} = \n\\bigcup\\_v {\\cal P}(\\Sigma^\\*)(\\text{eng}, v) | v \\in \\Pi(l, T)$. Furthermore, we will only consider the strings *t* ∈ T which occur more than Π(eng, *F*)/10 in the target-language verses of Π(eng, *F*) and *s* ∈ S which occur more than 2 times in the English verses of Π(*l*, *T*).\nMeasuring association\n---------------------\nGiven a set of strings *U* in a language *l*2, we want to find a string in language *l*1 that is associated with *U*. To this end, we use *χ*2 score to measure the degree of the association. Specifically, we divide the verse set into two subsets: verses containing *U* and verses not containing *U* in the Bible of *l*2, i.e., Π(*l*2, *U*) and ¬Π(*l*2, *U*). We then build a contingency table for each string candidate $t \\in {\\cal P}(\\Sigma^\\*)$ and *t* comes from language *l*1, as shown in Table [tab:cont]. After that, we compute the *χ*2 score for each string: the higher the score, the more associated the string in *l*1 is with the set of strings *U*. We then choose the string that has the highest *χ*2 score as a hit in language *l*1 for a set of strings *U* in language *l*2.\n| | in Π(*l*2, *U*) | in ¬Π(*l*2, *U*) |\n| --- | --- | --- |\n| *I*(*t*) | *n*00 | *n*01 |\n| ¬*I*(*t*) | *n*10 | *n*11 |\nAdding edges\n------------\nFor efficiency and stability reasons, in the actual implementation of the backward pass, does not add edges from a target node to all the strings in *S* that fulfill the criterion, i.e., the set of the identified associated source language strings, as shown in Algorithm [alg:bsnew] (line 9). Instead, we only add one edge only from each target node. This means that in each iteration of the backward pass, we will add new edges starting from the involved target nodes to a single *s*: $\\{((l,v,T),s)|(F, (l,v,T)) \\in {\\cal G} \\wedge s \\in v\\ \\wedge v \\in V\\}$. In this way, each target node that was previously connected with the focal concept *F* can only be connected to one source node only. By doing this, we find that some undesirable associations can be avoided.\nHyperparameters\n---------------\nWe have two hyperparameters in, i.e., (1) the maximum number of iterations of searching associated strings for a focal concept in each language: *M* and (2) the threshold *α* for the minimum coverage of the set of identified associated string *U*. We set *M* = 5 and *α* = .9 as default values for all involved computations. Based on preliminary experiments on ‘bird’ concept, we found that the number of associated strings usually will not go beyond 5. Moreover, when *M* is large, we might have an efficiency problem (more iterations for each language) when computing other focal concepts. Therefore, we set *M* = 5 to reduce the runtime of while not sacrificing the accuracy too much. As for the coverage threshold *α*, we conduct preliminary experiments with.85,.9 and.95 respectively on ‘bird’ concept. We found when coverage is small (<.9), the search stops when there are still possibly unidentified associated strings in the rest verses. If the coverage is too large (>.9), we found that some less related strings can be identified at the later stage of iterations for some languages. We should note that PBC can have verse-level misalignment problems, which means for some parallel verses ‘bird’ occurs in English but the target-language verse can be unrelated to ‘bird’ at all. Moreover, as we remove the parallel verses that we have covered in each iteration, the verses uncovered become smaller and smaller in each iteration. *χ*2 scores computed on later iterations can be not significant and multiple strings can have the same highest *χ*2 scores if they only appear in the uncovered verses with the same number of occurrences. Therefore, to ensure that finds enough strings while guaranteeing the quality of the associations between them with the search string, we set the coverage threshold *α* = .9.\n![Part of the document for annotation for one target language, i.e., aai (Arifama-Miniafia) The true positive and false positive verses are not shown here.](./figs/annotation_example.jpg \"fig:\")[fig:annotation]\n[fig:categories]\nDetails of Evaluation\n=====================\nSingle concept across all languages\n-----------------------------------\nIn the English version of the Bible, we find the ‘bird’ concept is often expressed by the following words/phrases: “bird(s)”, “fowl(s)”, “flying creature(s)” and “winged creature(s)”. Therefore, we use the following strings: `$bird`, `$fowl`, `$flying$creature` and `$winged$creature` to represent the ‘bird’ concept. We agree that this set of strings might not be optimal for other large-resource parallel datasets (if there are any). However, for PBC dataset, this set of strings could empirically cover the ‘bird’ concept. We perform our which includes the forward pass and backward pass for this ‘bird’ concept for all the languages in PBC. We perform three types of evaluations as follows based on the FP and BP hits:\n### ‘bird’ conceptualization in all languages\nWe provide a pdf document in which the statistics of each string identified in both forward pass and backward pass are shown. In addition, for each target-language string, a) two randomly sampled **True Positive** parallel verses, i.e., target-language verses that contain the identified strings and the parallel English verses contain ‘bird’; b) two randomly sampled **False Positive** parallel verses, i.e., target-language verses that contain the strings and the parallel English verses that do not contain ‘bird’, are shown. We also show three randomly sampled **False Negative** parallel verses, i.e., target-language verses that do not contain the strings and the parallel English verses that contain ‘bird’ concept. An example of part of the document for one language is shown in Figure [fig:annotation]. By checking the general patterns demonstrated in the document, we define four evaluation categories: **one-to-one**, **polysemy**, **ambiguity** and **failure**. Noticeably, our category **polysemy** and **ambiguity** do not directly correspond to the definition in linguistics, but reflect general patterns of the conceptualization. The classification of these two categories is based on the pattern of strings identified in the backward pass. More specifically, we classify the conceptualization pattern of a language as **polysemy**, if it shows one of the following patterns: (a) *hyponymy*, where we found strings such as `dove` and `sparrows`, which are hyponyms of ‘bird’. (b) *meronymy*, where we found strings like `$wings`, which is a meronym of ‘bird’. (c) *other related words*, where we found strings such as `$fly` and `$chirp`, which are apparently related to ‘bird’, but do not fit into any well-defined lexical semantic relation. The conceptualization pattern is classified as **ambiguity**, if the strings we found in the backward pass are not semantically related to ‘bird’ at all (such as `$new$` and `$kid$`), but nevertheless are deemed as highly associated with ‘bird’ by our algorithm. These cases are generally caused by having homonyms of ‘bird’ in the target language. In case the linguist annotator cannot be sure of the classification, consultation has been made with other experts to resolve these issues and find the most common agreement.\n| *l* | *P* | *T* | category |\n| --- | --- | --- | --- |\n| ify | ke:keq, qemayuq | `$sisit$` | **no overlap** |\n| ind | cewek, burung | `$burung`, `terbang$` | **overlap** |\n| akb | unggas | `$unggas$` | **match** |\n| afr | voël, vliegtuig | `voël` | **match** |\n| *l* | *P* | *T* | factor |\n| --- | --- | --- | --- |\n| uig | qush | (qush) | **script** |\n| mua | | `juu` | **transcription** |\n| lip | | | **prefix** |\n| mse | layra | `layagi` | **suffix** |\n| sbl | ma`nok | `manokmanok` | **reduplication** |\n### Translations compared with PanLex\nFor each language, the linguist annotator also checks the translation of “bird” provided by PanLex [4](#fn4). The translations are available in 1,137 languages out of the 1,331 languages in PBC where we found translations of the focal concept. We define the following four categories for the PanLex evaluation where *P* are the tranlations from PanLex and *T* are the FP hits (target-language strings).\n* **no translation**: *P* = ∅, i.e., PanLex gives no translation.\n* **no overlap**: *T* ∩ *P* = ∅, i.e., none of the FP hits is found in PanLex translations.\n* **overlap**: 0 < ∣*T* ∩ *P*∣ < ∣*T*∣, i.e., some but not all of the FP hits are found in PanLex translations.\n* **match**: ∣*T* ∩ *P*∣ = ∣*T*∣, i.e., all the FP hits can be found in PanLex translations. Note that we do not require all the translations in PanLex to be present in our set of target strings, since PanLex often gives a very long list of translations and our goal is to use PanLex translations to confirm the strings we identified.\nWe show examples for each category (except for **no translation**) in Table [tab:panlexexamples]. When deciding whether a translation from PanLex matches an FP hit, the linguist annotator does not only look for an exact match of strings but also takes the differences in scripts, transcriptions, and morphological forms into consideration. For languages with multiple writing systems, words are naturally transliterated into a unified script for them to be comparable. It has also been observed, that the same word can sometimes be transcribed differently in different sources, especially for low-resource languages that have no standard writing system. Therefore, “” in PanLex and “juu” in our FP hit will still be considered as a match. Furthermore, it is also possible that the PanLex translation uses a morphologically different form compared to our FP hit, such as dictionary form versus inflected form. Possible morphological processes such as affixation, reduplication, vowel mutation, vowel ellipsis, and metathesis have been taken into consideration. We show some examples of these factors in Table [tab:panlexnotexactmatch].\nWith careful examination, if the linguist annotator concludes that actually has found the same lexeme as the PanLex translation, and the difference between PanLex and our FP hit is merely attributed to the above-mentioned factors, they will still be considered as matches.\n| model | Coverage(g) | Coverage(a) | trans. per *l* |\n| --- | --- | --- | --- |\n| |.93 |.95 | 1.6 |\n| Eflomal 0 |.94 |.96 | 7.0 |\n| Eflomal 1 |.81 |.82 | 2.9 |\n| Eflomal 0.1 |.70 |.79 | 2.1 |\n### Translations compared with Eflomal\nWe also compare the coverage and the average number of translations proposed per language of and eflomal, a statistical word alignment tool, in Table [tab:birdresultsvseflomal]. We collect words that are aligned with one of the strings representing ‘bird’ in all verses where ‘bird’ occurs for eflomal baselines. The coverage means the fraction of verses containing ‘bird’ covered by the set of proposed translations. Global coverage denotes that we compute the coverage directly in all verses regardless of language while average coverage denotes that we first compute the coverage for each language and then average over all languages. We notice that, eflomal, without filtering any translations, obtains the highest global and average coverage, which can be regarded as the upper bound. However, the number of translations per language on average is so high: 7.0. After filtering some proposed translations by their frequencies (1 and 0.1), we observe a sudden drop in the coverage. This indicates that (1) eflomal can propose many wrong alignments and (2) some correct alignments have very small frequencies. Because of its word-level alignment nature, eflomal cannot take the possible morphological changes of the words into consideration at all. On the contrary, only proposes 1.6 translations on average while keeping the coverage very close to the upper bound, suggesting that can identify the strings (ngrams) that are most associated with the concept and alleviate the possible problems caused by, e.g., morphological changes, in many languages.\nSwadesh concepts\n----------------\nWe resort to NoRaRe [5](#fn5) to find the available translations of the 32 Swadesh concepts in 39 languages (NoRaRe covers). For each concept and each language, we store a triple indexed by the concept-language pair: \n < concept, language, translation(s) > \n We finally obtain 582 triples for evaluation (NoRaRe does not provide translations of a concept in all covered languages). We use *T* to represent the set of translations proposed by and *N* for NoRaRe translations (as ground-truth translations) in the triple. When judging whether a translation in *N* matches a translation in *T* generated by, we do the match leniently to allow for morphological changes. Specifically, if a translation in *N* is a substring of a translation in *T* generated by or the other way around, we regard it as a successful match. This is because *N* often provide the dictionary forms of the nouns but *T* are generated automatically based on the actual Bible verses where the nouns can change their suffixes or prefixes quite often depending on their roles in the verses.\nWe are especially interested in the triples in which our identified strings *T* do not match the ground-truth translations *R*, i.e., *T* ∩ *R* = ∅. We sampled 10 such triples (we provide *N* and *T* for each triple) in Table [tab:swadesherrors]. We notice that there are cases where the ground-truth translations *N* use different versions of transliterations. For example, `$andjing` vs. “anjing”, `$daoen` vs. “daun” and `$boelan` vs. “bulan” in Malay (msa). Moreover, there can be multiple equivalent translations for a concept, but *N* just lists one of them which is not used (or not identified) in PBC, e.g., ``\nUTF8gbsn颈\n” is a simpler but more formal translation of ‘neck’ but the *N* only lists ``\nUTF8gbsn脖子\n” in Chinese (zho); the concept ‘path’ can also be translated to “rout” and “sentier” in French (fra) but only “chemin” is given by the *N*. Therefore, we see that this evaluation compared with NoRaRe can actually underestimate the performance of our method.\nllll concept & *l* & *N* & *T* \n‘dog’ & msa & anjing & `$andjing` \n‘seed’ & cym & hedyn & `$had$`, `$heu` \n‘leaf’ & msa & daun & `$daoen` \n‘horn’ & est & ruupor & `sarve` \n‘mouth’ & tur & ağiz & `$ağzı`, `$ağız` \n‘neck’ & zho &\nUTF8gbsn脖子\n&\nUTF8gbsn颈\n \n‘moon’ & msa & bulan & `$boelan` \n‘water’ & cym & dŵr & `dwfr$`, `dyfr` \n‘rain’ & msa & hujan & `$hoedjan` \n‘path’ & fra & chemin & `$la$rout`, `$sentier` \n| Concepts | Concreteness | Stability |\n| --- | --- | --- |\n| ‘fish’ | 5.0 | 0.86 |\n| ‘bird’ | 5.0 | 0.68 |\n| ‘dog’ | 4.85 | 0.85 |\n| ‘tree’ | 5.0 | 0.64 |\n| ‘seed’ | 4.71 | 0.38 |\n| ‘leaf’ | 5.0 | 0.74 |\n| ‘root’ | 4.34 | 0.78 |\n| ‘flesh’ | 4.59 | 0.36 |\n| ‘blood’ | 4.86 | 0.69 |\n| ‘horn’ | 5.0 | 0.82 |\n| ‘hair’ | 4.97 | 0.77 |\n| ‘ear’ | 5.0 | 0.46 |\n| ‘mouth’ | 4.74 | 0.49 |\n| ‘tooth’ | 4.89 | 0.91 |\n| ‘tongue’ | 4.93 | 0.61 |\n| ‘foot’ | 4.9 | 0.7 |\n| ‘knee’ | 5.0 | 0.38 |\n| ‘belly’ | 4.8 | 0.4 |\n| ‘neck’ | 5.0 | 0.72 |\n| ‘breast’ | 4.89 | 0.65 |\n| ‘sun’ | 4.83 | 0.49 |\n| ‘moon’ | 4.9 | 0.48 |\n| ‘star’ | 4.69 | 0.87 |\n| ‘water’ | 5.0 | 0.48 |\n| ‘rain’ | 4.97 | 0.68 |\n| ‘stone’ | 4.72 | 0.71 |\n| ‘cloud’ | 4.54 | 0.68 |\n| ‘smoke’ | 4.96 | 0.57 |\n| ‘path’ | 4.41 | 0.35 |\n| ‘mountain’ | 4.96 | 0.64 |\n| ‘white’ | 3.89 | 0.77 |\n| ‘night’ | 4.52 | 0.68 |\n| Concepts | Concreteness | Stability |\n| --- | --- | --- |\n| ‘babe’ | 3.67 | 0.59 |\n| ‘hypocrit’ | 2.43 | 0.81 |\n| ‘soldier’ | 4.72 | 0.49 |\n| ‘scroll’ | 4.11 | 0.57 |\n| ‘demon’ | 3.32 | 0.45 |\n| ‘boat’ | 4.93 | 0.71 |\n| ‘olive’ | 4.9 | 0.8 |\n| ‘prayer’ | 3.28 | 0.32 |\n| ‘mercy’ | 1.57 | 0.29 |\n| ‘trumpet’ | 4.86 | 0.83 |\n| ‘angel’ | 3.82 | 0.88 |\n| ‘prison’ | 4.68 | 0.62 |\n| ‘savior’ | 3.04 | 0.49 |\n| ‘tomb’ | 4.73 | 0.61 |\n| ‘husband’ | 4.11 | 0.47 |\n| ‘bride’ | 4.63 | 0.69 |\n| ‘talent’ | 2.19 | 0.83 |\n| ‘peace’ | 1.62 | 0.72 |\n| ‘secret’ | 2.19 | 0.57 |\n| ‘faith’ | 1.63 | 0.59 |\n| ‘woe’ | 1.96 | 0.8 |\n| ‘throne’ | 4.64 | 0.62 |\n| ‘wisdom’ | 1.53 | 0.54 |\n| ‘disciple’ | 3.29 | 0.73 |\n| ‘obeisance’ | NA | 0.37 |\n| ‘truth’ | 1.96 | 0.4 |\n| ‘memor’ | 2.83 | 0.53 |\n| ‘governor’ | 4.07 | 0.52 |\n| ‘poor’ | 2.7 | 0.63 |\n| ‘blind’ | 4.03 | 0.77 |\n| ‘spiritual’ | 1.79 | 0.33 |\n| ‘justice’ | 1.45 | 0.34 |\n| ‘courage’ | 1.52 | 0.53 |\n| ‘purpose’ | 1.52 | 0.3 |\n| ‘generation’ | 1.96 | 0.56 |\n| ‘contrary’ | 1.56 | 0.46 |\n| ‘prophesy’ | 2.11 | 0.41 |\n| ‘decision’ | 2.19 | 0.36 |\n| ‘request’ | 2.59 | 0.32 |\n| ‘weakness’ | 2.59 | 0.55 |\n| ‘journey’ | 2.57 | 0.39 |\n| ‘public’ | 2.57 | 0.23 |\n| ‘appearance’ | 2.57 | 0.55 |\n| ‘expression’ | 2.54 | 0.51 |\n| ‘marriage’ | 2.51 | 0.51 |\n| ‘wrath’ | 2.42 | 0.4 |\n| ‘trouble’ | 2.25 | 0.45 |\n| ‘promise’ | 2.09 | 0.46 |\n| ‘power’ | 2.04 | 0.41 |\n| ‘pleasure’ | 2.04 | 0.35 |\n| ‘thought’ | 1.97 | 0.39 |\nInfrastructure & environment\n============================\nWe ran all our computational experiments on a CPU server with 48 cores and 1024 GB of memory. We used Python 3.6[6](#fn6) throughout our implementation of and for visualizations. Specifically, for fundamental scientific computing (e.g., computing *χ*2 scores), we used NumPy[7](#fn7), SciPy[8](#fn8) and scikit-learn[9](#fn9) packages. For visualization, we used NetworkX[10](#fn10)(mainly for the crosslingual semantic fields) and Matplotlib[11](#fn11) packages.\nCrosslingual semantic fields\n============================\n[fig:semanticfield1]\n[fig:semanticfield2]\n[fig:semanticfield3]\n[fig:semanticfield4]\n[fig:semanticfield5]\nFurther analysis regarding language similarity\n==============================================\n[htbp] [fig:clustering5183]\nlcccccccc setting & global & atla & aust & indo & guin & otom & sino \nSwadesh32 & 1280 & 222 & 215 & 91 & 87 & 76 & 68 \nBible51 & 1264 & 219 & 210 & 90 & 85 & 76 & 67 \nAll83 & 1263 & 219 & 210 & 90 & 85 & 76 & 67 \nlcccccccc #neighbors & global & Papunesia & Africa & Eurasia & North America & South America & Australia \nk=1 & 0.51 & 0.51 & 0.52 & 0.64 & 0.52 & 0.35 & 0.00 \nk=2 & 0.28 & 0.27 & 0.31 & 0.47 & 0.27 & 0.09 & 0.00 \nk=3 & 0.45 & 0.44 & 0.51 & 0.62 & 0.43 & 0.22 & 0.00 \nk=4 & 0.53 & 0.52 & 0.60 & 0.69 & 0.51 & 0.25 & 0.00 \nk=5 & 0.55 & 0.54 & 0.61 & 0.72 & 0.57 & 0.28 & 0.09 \nk=6 & 0.56 & 0.55 & 0.62 & 0.75 & 0.57 & 0.26 & 0.09 \nk=7 & 0.56 & 0.55 & 0.63 & 0.74 & 0.56 & 0.24 & 0.09 \nk=8 & 0.57 & 0.56 & 0.64 & 0.76 & 0.56 & 0.27 & 0.00 \nk=9 & 0.57 & 0.54 & 0.66 & 0.77 & 0.59 & 0.26 & 0.00 \nk=10 & 0.58 & 0.55 & 0.65 & 0.79 & 0.62 & 0.22 & 0.00 \nlcccccccc #neighbors & global & Papunesia & Africa & Eurasia & North America & South America & Australia \nk=1 & 0.49 & 0.54 & 0.57 & 0.63 & 0.28 & 0.30 & 0.00 \nk=2 & 0.24 & 0.25 & 0.33 & 0.42 & 0.06 & 0.08 & 0.00 \nk=3 & 0.41 & 0.43 & 0.59 & 0.62 & 0.13 & 0.09 & 0.00 \nk=4 & 0.48 & 0.52 & 0.66 & 0.70 & 0.20 & 0.14 & 0.00 \nk=5 & 0.51 & 0.55 & 0.70 & 0.72 & 0.18 & 0.18 & 0.00 \nk=6 & 0.50 & 0.53 & 0.72 & 0.70 & 0.18 & 0.15 & 0.00 \nk=7 & 0.50 & 0.53 & 0.71 & 0.70 & 0.18 & 0.16 & 0.00 \nk=8 & 0.49 & 0.51 & 0.71 & 0.69 & 0.14 & 0.15 & 0.00 \nk=9 & 0.48 & 0.48 & 0.73 & 0.71 & 0.12 & 0.12 & 0.00 \nk=10 & 0.49 & 0.50 & 0.74 & 0.71 & 0.14 & 0.12 & 0.00 \nlcccccccc #neighbors & global & Papunesia & Africa & Eurasia & North America & South America & Australia \nk=1 & 0.54 & 0.59 & 0.65 & 0.68 & 0.38 & 0.25 & 0.00 \nk=2 & 0.33 & 0.35 & 0.41 & 0.53 & 0.16 & 0.09 & 0.00 \nk=3 & 0.51 & 0.51 & 0.69 & 0.71 & 0.27 & 0.17 & 0.00 \nk=4 & 0.56 & 0.57 & 0.75 & 0.75 & 0.32 & 0.21 & 0.00 \nk=5 & 0.58 & 0.59 & 0.77 & 0.78 & 0.34 & 0.22 & 0.00 \nk=6 & 0.58 & 0.60 & 0.77 & 0.79 & 0.35 & 0.17 & 0.00 \nk=7 & 0.58 & 0.60 & 0.79 & 0.81 & 0.34 & 0.15 & 0.00 \nk=8 & 0.57 & 0.59 & 0.77 & 0.81 & 0.33 & 0.16 & 0.00 \nk=9 & 0.58 & 0.59 & 0.78 & 0.81 & 0.32 & 0.18 & 0.00 \nk=10 & 0.58 & 0.60 & 0.79 & 0.80 & 0.32 & 0.17 & 0.00 \n| | | |\n| --- | --- | --- |\n| | zho | `$mouth$`, `$mouths$`, `$entrance`, `alat`, `$ford` |\n| | bod | `$mouth$`, `$mouths$`, `$prais` |\n| [1ex] | zho | `$neck$`, `$necks$`, `$stiff-necked` |\n| | bod | `$neck$`, `$necks$`, `$obstina`, `$stubbornness$` |\n| [1ex] | zho | `$tree$`, `$trees$`, `$vine$`, `-tree$`, `$boughs$` |\n| | bod | `$tree$`, `$trees$`, `$chariot`, `wood`, `$cedar`, `igs$`, `$timber` |\n| [1ex] | zho | `$horn$`, `$horns$`, `$corner` |\n| | bod | `$horn$`, `$horns$`, `$trumpet`, `$fathering$` |\n| | | |\n| --- | --- | --- |\n| | arb | `$fish$`, `$fishes$` |\n| | pes | `$fishes$`, `$fish$`, `$fish` |\n| | tur | `$fish$`, `$fishes$`, `$fish`, `$honey` |\n| | msa | `$fishes$`, `$fish$` |\n| | ind | `$fish$`, `$fishes$` |\n| [1ex] | arb | `$stars$`, `$star$` |\n| | pes | `$stars$`, `$star$` |\n| | tur | `$stars$`, `$star$` |\n| | msa | `$stars$`, `$star$` |\n| | ind | `$stars$`, `$star$` |\n| [1ex] | arb | `$blood$`, `$bloodguilt$` |\n| | pes | `$blood$` |\n| | tur | `$blood$`, `$bloods`, `$bloodguilt$` |\n| | msa | `$blood$` |\n| | ind | `$blood$`, `$blood` |\n| [1ex] | arb | `$tongue$`, `$tongues$` |\n| | pes | `$tongue$`, `$tongues$` |\n| | tur | `$tongue$`, `$tongues$`, `$request`, `$language` |\n| | msa | `$tongues$`, `$tongue$`, `ebrew$` |\n| | ind | `$tongue$`, `$tongues$`, `$language` |\n| [1ex] | arb | `$birds$`, `$bird$`, `$flying$`, `$fowls$` |\n| | pes | `$birds$`, `$bird$` |\n| | tur | `$birds$`, `$bird$`, `$fowl` |\n| | msa | `$birds$`, `$bird$`, `dove`, `$sparrows$`, `$eagle` |\n| | ind | `$birds$`, `$bird$`, `$eagle`, `$turtledove`, `$ostrich`, `$raven`, `$bird` |\n| | | |\n| --- | --- | --- |\n| | eng | `$blood$` |\n| | spa | `$blood$` |\n| | ell | `$blood$` |\n| | rus | `$blood$`, `$blood` |\n| | tgl | `$blood$` |\n| | swh | `$blood$`, `$blood` |\n| | hye | `$blood$`, `$blood` |\n| [1ex] | eng | `$tongue$`, `$tongues$` |\n| | spa | `$tongue$`, `$tongues$`, `$language` |\n| | ell | `$tongue$`, `$tongues$`, `$language` |\n| | rus | `$tongue$`, `$tongues$`, `$language` |\n| | tgl | `$tongue$`, `$tongues$`, `$language` |\n| | swh | `$tongue$`, `$tongues$`, `$language` |\n| | hye | `$tongue$`, `$tongues$`, `$language` |\n| [1ex] | eng | `$birds$`, `$bird$` |\n| | spa | `$birds$`, `$bird$`, `$fowls$` |\n| | ell | `$birds$`, `$bird$`, `$bird` |\n| | rus | `$birds$`, `$bird$`, `$fowl`, `$bird` |\n| | tgl | `$birds$`, `$bird$`, `$fowl` |\n| | swh | `$birds$`, `$bird$`, `$fowl` |\n| | hye | `$birds$`, `$bird$`, `$flying$`, `$fowl` |\n| | | |\n| --- | --- | --- |\n| | eng | `$blood$` |\n| | deu | `$blood$` |\n| | fra | `$blood$` |\n| | jpn | `$blood$`, `$blood` |\n| | kor | `$blood$`, `$escap`, `$skin$`, `$airs$`, `$pip`, `$flut` |\n| | zho | `$blood$`, `$blood` |\n| [1ex] | eng | `$tongue$`, `$tongues$` |\n| | deu | `$tongue$`, `$tongues$` |\n| | fra | `$tongue$`, `$tongues$`, `$language` |\n| | jpn | `$tongue$`, `$tongues$` |\n| | kor | `$tongue$`, `$tongues$`, `$language` |\n| | zho | `$tongue$`, `$tongues$`, `$language` |\n| [1ex] | eng | `$mouth$`, `$mouths$` |\n| | deu | `$mouth$`, `$mouths$` |\n| | fra | `$mouth$`, `$mouths$` |\n| | jpn | `$mouth$`, `$mouths$`, `$entrance`, `$kiss`, `$whistl`, `$doorkeeper`, `$contention` |\n| | kor | `$mouth$`, `$mouths$`, `$entrance`, `$lip`, `$clothe`, `$kiss`, `$overla` |\n| | zho | `$mouth$`, `$mouths$`, `$entrance`, `alat`, `$ford` |\n| | | |\n| --- | --- | --- |\n| | spa | `$birds$`, `$bird$`, `$fowls$` |\n| | tgl | `$birds$`, `$bird$`, `$fowl` |\n| | ceb | `$birds$`, `$bird$`, `$fowl` |\n| | hil | `$birds$`, `$bird$`, `$sparrows$` |\n| [1ex] | spa | `$ear$`, `$ears$`, `$heard$` |\n| | tgl | `$ears$`, `$ear$`, `$listen$`, `$hearing$` |\n| | ceb | `$ears$`, `$ear$`, `$hear$`, `$hearing$` |\n| | hil | `$ear$`, `$ears$`, `$hear` |\n| [1ex] | spa | `$tongue$`, `$tongues$`, `$language` |\n| | tgl | `$tongue$`, `$tongues$`, `$language` |\n| | ceb | `$tongue$`, `$tongues$`, `$language` |\n| | hil | `$tongue$`, `$tongues$` |\n| | | |\n| --- | --- | --- |\n| | yor | `$tree$`, `$trees$`, `wood`, `$stake`, `$frankincense$`, `$thornbush`, `$palm-tree$` |\n| | ibo | `$tree$`, `$trees$`, `$pole`, `wood`, `$impal`, `$stake`, `$panel` |\n| | mcn | `$tree$`, `$trees$`, `wood`, `$stake`, `$impale`, `$cedar`, `$timber` |\n| | twi | `$tree$`, `$trees$`, `$wood`, `$panel$`, `$pole`, `$figs$`, `$timber` |\n| [1ex] | yor | `$hair$`, `$hairs$`, `$wool$` |\n| | ibo | `$hair$`, `$hairs$`, `$wool`, `$shear`, `$beard` |\n| | mcn | `$hair$`, `$hairs$`, `$wool$`, `$shave`, `$baldness$`, `$shear`, `goat` |\n| | twi | `$hair$`, `$hairs$`, `$beard`, `$shave`, `$head$`, `$wool` |\n| [1ex] | yor | `$mouth$`, `$mouths$`, `$entrance`, `$kiss`, `$palate$`, `$marvel`, `$suckling` |\n| | ibo | `$mouth$`, `$mouths$`, `$gate`, `$entrance`, `$lip`, `curse`, `$precious$` |\n| | mcn | `$mouth$`, `$mouths$`, `$lips$`, `fulfill`, `$denie`, `$disown`, `$entrance` |\n| | twi | `$mouth$`, `$mouths$`, `$gat`, `$collect`, `$lip`, `$entrance`, `$registered$` |\n| | | |\n| --- | --- | --- |\n| | cak | `$tree$`, `$trees$`, `wood`, `$pole`, `$cedar`, `$figs$`, `$palm-tree$` |\n| | kjb | `$tree$`, `$trees$`, `$cedar`, `$panel`, `$wood`, `$figs$`, `$pole` |\n| | tzj | `$tree$`, `$trees$` |\n| [1ex] | cak | `$seed$`, `$seeds$`, `braham`, `$vine`, `$sow`, `fruit`, `$harvest` |\n| | kjb | `$seed$`, `$seeds$`, `braham`, `$garden`, `fruits$`, `$vine`, `$harvest` |\n| | tzj | `$seed$`, `$seeds$`, `$sow`, `$harvest$`, `fruit` |\n| [1ex] | cak | `$knees$`, `$bow`, `$worship`, `$trembl`, `fell$` |\n| | kjb | `$knees$`, `$knee$`, `$obeisance$`, `$worship`, `$bow`, `$fell$` |\n| | tzj | `$knees$`, `$worship`, `$obeisance$`, `$fell$` |\nccll & & \n & jpn &\nUTF8bsmi口\n(mouth, opening, entrance) \n& kor &\nUTF8mj구 (口)\n(entrance, gate, mouth) \n& zho &\nUTF8bsmi口 (mouth, gate, entrance), 嘴 (mouth, lips)\n \n& fra & bouche (mouth) \n[1ex] & spa & lengua (tongue, language) \n& tgl & dilà (tongue, language), wikà (tongue, language) \n& ceb & dila (tongue), pinulongan (tongue, language) \n& hil & dilà (tongue), dilâ (tongue) \n& msa & lidah (tongue), oojoo leeda (tongue) \n[1ex]\n---\n1. We release our code at [https://github.com/yihongL1U/conceptualizer]( https://github.com/yihongL1U/conceptualizer)[↩](#fnref1)\n2. For some focal concepts that are less divergent, e.g.,, we obtain fewer than 99 dimensions[↩](#fnref2)\n3. https://iso639-3.sil.org/[↩](#fnref3)\n4. https://translate.panlex.org/[↩](#fnref4)\n5. https://norare.clld.org/[↩](#fnref5)\n6. https://www.python.org/[↩](#fnref6)\n7. https://numpy.org/[↩](#fnref7)\n8. https://scipy.org/[↩](#fnref8)\n9. https://scikit-learn.org/stable/[↩](#fnref9)\n10. https://networkx.org/[↩](#fnref10)\n11. https://matplotlib.org/[↩](#fnref11)\n\nA Crosslingual Investigation of Conceptualization in 1335 Languages\n===================================================================\n=1\nLanguages differ in how they divide up the world into concepts and words; e.g., in contrast to English, Swahili has a single concept for and. We investigate these differences in conceptualization across 1,335 languages by aligning concepts in a parallel corpus. To this end, we propose, a method that creates a bipartite directed alignment graph between source language concepts and sets of target language strings. In a detailed linguistic analysis across all languages for one concept (‘bird’) and an evaluation on gold standard data for 32 Swadesh concepts, we show that has good alignment accuracy. We demonstrate the potential of research on conceptualization in NLP with two experiments. (1) We define crosslingual stability of a concept as the degree to which it has 1-1 correspondences across languages, and show that concreteness predicts stability. (2) We represent each language by its conceptualization pattern for 83 concepts, and define a similarity measure on these representations. The resulting measure for the conceptual similarity between two languages is complementary to standard genealogical, typological, and surface similarity measures. For four out of six language families, we can assign languages to their correct family based on conceptual similarity with accuracies between 54% and 87%.[1](#fn1)\nIntroduction\n============\nLanguages differ in how they divide up the world into concepts and words. The Swahili word ‘tumbo’ unites the meanings of the English words ‘belly’ and ‘womb’. Therefore, English forces its speakers to differentiate between the general body region “front part of the human trunk below the ribs” and one particular organ within it (the womb) whereas Swahili does not. Similarly, Yoruba ‘irun’ refers to both hair and wool. Again, English speakers must make a distinction whereas Yoruba has a single hair concept that includes the meaning.\n [fig:pipeline]\nWhile studies have looked at conceptualization within different languages, we present a crosslingual study that directly compares conceptualization in 1,335 languages. The empirical basis are word and ngram correspondences in the **P**arallel **B**ible **C**orpus (PBC, ). We introduce, a method that reliably aligns a set of 83 concepts across all PBC languages. The 83 concepts are partly chosen to be well represented in the Bible, and partly from Swadesh 100. The alignments are formalized as a bipartite graph between English (the source) and the target languages.\nThe simple idea underlying – illustrated in Figure [fig:pipeline] – is that, starting with one of the 83 concepts in English as the focal concept (the search query), we can identify divergent conceptualizations by first searching for target ngrams highly associated with the focal concept, and then searching for English ngrams highly correlated with the target ngrams we found. If the English ngrams correspond to the original focal concept, then the conceptualizations do not diverge. In contrast, take the example of divergence described above: we start with the focal concept, find Yoruba ‘irun’ and then two English concepts, not one, that are highly associated with ‘irun’: and. This indicates that English and Yoruba conceptualizations diverge for.\nOur main contribution is that we present the first empirical study of crosslingual conceptualization that grounds the semantics of concepts directly in contexts – the sentences of the parallel corpus. This ensures that our work is based on identical (or at least very similar) meanings across all 1,335 languages we investigate. For example, verse Matthew 9:7 has the same meaning in English: “Then the man got up and went home.”, in Chinese: ``\nUTF8bsmi那個人就起來，回家去了。\n” and in each of the other 1,333 languages. Such a direct grounding in meaning across a large set of languages has not previously been achieved in work on conceptualization in theoretical or computational linguistics.\nIn addition, we make the following contributions. (i) We propose, an alignment method specifically designed for concept alignment, that operates on the level of ngrams and ngram sets. (ii) We conduct an evaluation of for the concept in all 1,335 languages. The result is a broad characterization of how the conceptualization of bird varies across the languages of the world. Out of 1,335 languages, only fails 15 times (due to data sparseness) for. (iii) We evaluate for 32 Swadesh concepts on a subset of 39 languages for which translation resources exist and demonstrate good performance. (iv) Using the ratings provided by, we give evidence that concreteness (i.e., the degree to which a concept refers to a perceptible entity) causes a concept to be more stable across languages: concrete concepts are more likely to have one-to-one mappings than abstract concepts. (v) We propose a new measure of language similarity. Since we have aligned concepts across languages, we can compute measures of how similar the conceptualization of two languages is. We show that this gives good results and is complementary to genealogical, typological and surface similarity measures that are commonly used. For example, Madagascar’s Plateau Malagasy is conceptually similar to geographically distant typological relatives like Hawaiian, but also to typologically distant “areal neighbors” like Mwani and Koti. For four out of six language families, based on conceptual similarity, we can assign languages to their correct family with between 54% and 87% accuracy.\nRelated Work\n============\nIn linguistics, conceptualization has been studied empirically with regards to crosslingual polysemy or colexification as well as areal and cultural influences on concept similarity. Most of this work is based on human annotations, such as CLICS, a database of colexification. However, the coverage of such resources in terms of concepts included, especially for some low-resource languages, is low. Therefore we explore the use of an unannotated broad-coverage parallel corpus as an alternative. Expanding this work to many languages is important to the extent that we accept some (weak) form of linguistic relativity, i.e., the hypothesis that language structure (including conceptualization) influences cognition and perception.\nMethodologically, our work is closely related to who explores colexification through PBC. He targets specific colexification pairs and investigates their geographical distribution using word alignments. In comparison, our method allows us to identify alignments beyond the word level and therefore richer associations among concepts are obtained. Our proposed method is also close to semantic mirrors, a method to explore semantic relations using translational data. The authors focus on an English-Norwegian lemmatized parallel corpus; in contrast, we investigate 1,335 languages, most of which are low-resource and for many of which lemmatization is not available. In addition, this paper is related to recent work that uses PBC to investigate the typology of tense, train massive multilingual embeddings, extract multilingual named entities, find case markers in a multilingual setting and learn language embeddings containing typological features.\nLike, analyzed the semantic similarity of concepts across languages (mainly European ones). But they use pretrained word embeddings, which are not available in high enough quality for most of the low-resource languages we cover in this work.\nComputational criteria for language similarity have been taken from typology, morphology and language-model surface similarity. We propose a new similarity measure, based on conceptualization, with complementary strengths and weaknesses.\nThere is a large body of work on statistical and neural word alignment; recent papers with extensive discussion of this subfield include. We show below that the standard alignment method Eflomal does not work well for our problem, i.e., for identifying high-accuracy associations between concepts.\nMethodology\n===========\nData\n----\nWe work with the Parallel Bible Corpus (PBC, ). We use 1,335 Bible translations from PBC, each from a different language as identified by its ISO 639-3 code. For most languages, PBC only covers the New Testament (NT) ( ≈ 7,900 verses). For a few hundred, it covers both NT and Hebrew Bible ( ≈ 30,000 verses). See for details of the PBC corpus.\nFrom Swadesh 100, a set of 100 basic universal concepts, we select the 32 concepts that occur with frequency 5 < *f* ≤ 500 in both NT and Hebrew Bible. We call the resulting set of 32 concepts **Swadesh32**. We also select **Bible51** from the Bible, a set of 51 concepts that are of interest for crosslingual comparison. Notably, we include abstract concepts like that are missing from Swadesh32. See for concept selection details.\n**Bipartite graph.** We formalize the concept alignment graph as a directed bipartite graph. With Σ denoting the alphabet, let ${\\cal S} \\subset {\\cal P}(\\Sigma^\\*)$ be the set of source nodes, each corresponding to a concept, represented as a set of strings from the source language; e.g., {`$belly$`, `$bellies$`} for, where `$` denotes the word boundary. In this paper, we always use English as the source language. With Λ denoting the set of languages and Π the set of verses, let ${\\cal T} \\subset \\Lambda \\times \\Pi \\times {\\cal P}(\\Sigma^\\*)$ be the set of target nodes, each corresponding to a triple of target language *l*, Bible verse *v* and a set of strings from language *l*, one of them occurring in *v*. We represent the concept correspondences as a directed bipartite Graph ${\\cal G} \\subset\n{\\cal S} \\times {\\cal T} \\cup\n{\\cal T} \\times {\\cal S}$, as shown in. See for method details. gives our notation. The reason for our asymmetric design of the graph (concept *types* on the source side, concept *tokens* occurring in context on the target side) is that we want to track how much evidence there is for a concept-concept correspondence. The more edges there are in the graph, the more reliable the correspondence is.\nll\n7cm\n&\n8cm\n**Association for alignment.** We can represent a source concept (e.g., {`$belly$`, `$bellies$`}) as the set of verses *V* in which it occurs. In contrast to standard alignment algorithms, we exhaustively search all strings *t* of the target language *l* for high correlation with *V*. For example, we search for the French string *t* that has the highest correlation with the verses that contain {`$belly$`, `$bellies$`}; the result is *t*=. This means that we are not limited to knowing what the relevant (tokenization) units are in advance, which is not possible for all 1,335 languages. We use the *χ*2 score *χ*2(*l*, *t*, *V*) as a measure of correlation: we test, for all *t*, whether the two categorical variables *t* ∈ *v* (short for: *t* is a substring of verse *v* in language *l*) and *v* ∈ *V* are independent. We select the *t* with the highest score.\n**Termination.** For a query string *q*, e.g.,, occurring in verses *V*, we want to find a set *U* of highly associated ngrams in the target language *l* that covers all of *V*. Because of noise, translation errors, nonliteral language etc., this is often impossible. We therefore terminate the search for additional target strings when COVERAGE(*l*, *U*, *V*) ≥ *α* where we set *α* = .9 and define:\n \n$$\\begin{aligned}\n \\mathindent\\mathindent\\mbox{COVERAGE}(l,U,V) = \\frac{|\\Pi(l,U)\\cap V|}{|V|}\\end{aligned}$$ \ni.e., the fraction of *V* covered by the strings in *U*.\n**Graph induction.** shows that consists of a forward pass (FP, Algorithm [alg:fs]) that adds edges $e\n\\in {\\cal S} \\times {\\cal T}$ and a backward pass (BP, Algorithm [alg:bsnew]) that adds edges $e \\in {\\cal T} \\times {\\cal S}$ to ${\\cal\nG}$. FP and BP are essentially the same. To abstract from the direction, we will use the terms query language and retrieval language. In FP (resp. BP), the query language is the source (resp. target) language and the retrieval language is the target (resp. source) language.\n* Let *q* be the query string from the query language.\n* The set *R* holds retrieval language strings that are highly associated with *q*. *R* is initially empty. *R* is *T* (a set of target strings) or *S* (a set of English source strings) in the algorithms.\n* In each iteration, we find the retrieval language string *r* with the highest association to those verses containing the query *q* that are not yet covered by *R*.\n* We terminate when coverage by *R* (of verses containing *q*) exceeds the threshold *α*.\n* We return all edges that go from a query language node that contains *q* to a retrieval language node that contains a string from *R*.\nThe formal description in is slightly more complex because the query *q* is not a single string but a set. But this extension is straightforward. We now explain the formal description in.\nWe invoke FP and BP for all pairs (focal concept *F*, target language *l*) and merge the result with $\\cal G$ for each invocation. Writing PASS for FP or BP:\n \n$$\\begin{aligned}\n{\\cal G} \\gets \n{\\cal G} \\cup \\mbox{PASS}(F,l)\\end{aligned}$$ \nFor the following description of the algorithms, we write *s* ∈ *v* for “string *s* (in language *l*) is a substring of (the language *l* version of) verse *v*”. For brevity, we describe FP (Algorithm 1) [and describe BP (Algorithm 2) in square brackets]. Line 1: *T* [*S*] collects target [source] strings. Line 2: *M* is the maximum number of iterations; we set *M* = 5. Line 3: *V* is the set of verses that contain a string in *F* [were linked by an edge from *F* in FP], but are not yet covered by *T* [*S*]. Line 4: We save the result for *i* = 1 (or *T* = ∅ [*S* = ∅]) in *V*1, the base set of verses. Line 5: If the coverage that *T* [*S*] has of *V*1 exceeds a threshold *α*, we terminate; we set *α* = .9. Line 6: We find the target string *t* [source string *s*] that is most associated with *V*, ignoring target [source] string candidates already covered. Line 7: *t* [*s*] is added to *T* [*S*]. Line 9: In FP, we return a set of new edges that start at the focal concept *F* and end at a target node (*l*, *v*, *T*) whose verse *v* contains a string *t* from *T*. Line 9: In BP, we return a set of new edges that start at a target node (*l*, *v*, *T*) that was connected to *F* in FP and end at an *S*ʹ that contains a highly associated source string *s* (i.e., *s* ∈ *S*) in *v*.\nEvaluation\n==========\nSingle concept across all languages\n-----------------------------------\nWe first evaluate how well our method performs at identifying associated concepts across the highly diverse set of languages we cover. Since there is no appropriate broad-coverage high-quality resource, this requires an expensive manual analysis by a linguist. We can therefore only perform it for one concept in this paper. We choose the focal concept, defined as {`$bird`, `$fowl`, `$flying$creature`, `$winged$creature`}. For each language *l*, we analyze the hits we get for in *l*, primarily by looking at its BP hits in English, i.e., the English strings that are proposed in BP by running on. Defining *R* as the set of verses in which BP hits occur and *B* as the set of verses in which ‘bird’ occurs, we use four evaluation categories. (1) **one-to-one**. *R* ≈ *B*. In detail: ∣*R* − *B*∣ < .1∣*B*∣ and *R* − *B* does not contain plausible additional hits. (2) **polysemy**. *R* ⊃ *B* and *R* − *B* consists of verses with concepts closely related to, e.g.,,. (3) **ambiguity**. *R* − *B* contains verses in which neither ’bird’ nor closely related meanings occur. However, there is a second “non-bird” meaning of the BP hits; e.g., for Adioukrou the FP hit is “r” and the BP hits correspond to two clusters, a cluster and a cluster. (4) **failure**. *R* − *B* or *B* − *R* is large and this cannot be attributed to polysemy or (simple) ambiguity. See for details. (top) shows that found the translation of in almost all languages where we count the **ambiguity** case (e.g., Adioukrou “r” meaning both and ) as a success. The search failed for 4 languages (4 = 1335 − 1331) for which we have no verse that contains in English and 11 languages for many of which the number of verses was small. Thus, requires a large enough parallel corpus for good performance.\ncccc|c **1-1** & **polysemy** & **ambiguity** & **failure**& **total** \n687 & 579 & 54 & 11 & 1331 \n**match** & **overlap** & **no overlap** & **no translation**& **total** \n488 & 192 & 457 & 194 & 1331 \nWe also evaluate on PanLex, <http://panlex.org>. Defining *P* as the translations from PanLex and *T* as the FP hits for, we use the following four categories. (1) PanLex gives **no translation**. *P* = ∅. (2) **no overlap**. *P* ∩ *T* = ∅. (3) **overlap**. 0 < ∣*P* ∩ *T*∣ < ∣*T*∣. (4) **match**. ∣*P* ∩ *T*∣ = ∣*T*∣. See for details. (bottom) shows that for PanLex languages, performs well on ≈ 60%: (488 + 192)/(488 + 192 + 457). In a qualitative analysis, we found four reasons for the 457 **no overlap** cases. (i) A language has a very small corpus in PBC. (Sparseness was also the reason for failure in, top). (ii) did find correct translations of, but they are missing from PanLex. (iii) There is a dialect/variety mismatch Bible vs PanLex (no occurrence of the PanLex translation in our corpus). (iv) PanLex incorrectly translates through an intermediate language. For example, since PanLex has no direct translation of English to Chorote Iyowujwa, it goes through Gimi ‘nimi’ (which means both and ) and returns Chorote Iyowujwa ‘inxla7a’. But ‘inxla7a’ only means. Another example is that PanLex translates as ‘San’ instead of the correct ‘nghoq’ for Achang. Thus, PanLex translations through the intermediate mechanism are unreliable while our FP hit can find the correct translation.\nTaking the two evaluations together (manual analysis of BP hits and comparison of FP hits to PanLex translations), we interpret the results as indicating that reliably finds the correct translation of the focal concept, but can fail in case of data sparseness.\nlrrrr model & & & & \n& 87.21 & 84.88 & 89.69 & 1.03 \nEflomal 0 & 89.52 & 87.80 & 91.23 & 10.42 \nEflomal 1 & 86.98 & 84.88 & 89.18 & 4.50 \nEflomal 0.1 & 78.68 & 76.12 & 81.44 & 1.07 \nSwadesh concepts\n----------------\nWe next evaluate on Swadesh32 (). indicates that PanLex quality is low for many languages. We therefore use NoRaRe, <http://norare.clld.org>. We use all 582 concept-language pairs for which NoRaRe gives a translation. For a concept-language pair, let *T* be the proposed translations (from or Eflomal) and *N* gold (from NoRaRe). Then we compute recall as ∣*T* ∩ *N*∣/∣*N*∣. We match two ngrams if one is a substring of the other; e.g., is correct for. For Eflomal, we set *T* to the set of target language words aligned with one of the focal concept words (e.g., {`$belly$`, `$bellies$`}). Eflomal 0, 1, 0.1 denotes that we only keep translations whose frequency is > 0, > 1 and > .1Π(*l*, *F*), respectively.\nshows that ’s Swadesh32 translations have high recall (roughly 85% and higher, depending on the measure), with few false positives (1.03). For Eflomal, however, as we restrict matches to high-precision matches (i.e., going from 0 to 1 and to.1), both recall and false positives (FP) drop. Our interpretation is that the alignments obtained by Eflomal are noisy: Eflomal misaligns the focal concept with many irrelevant words. In contrast to, Eflomal offers no good tradeoff. This validates that we use instead of standard aligners like Eflomal. Most importantly, the evaluation on NoRaRe shows that has high recall and produces few false positives, which are prerequisites for further reliable exploration/analysis. See for details of the evaluation (including an additional experiment in terms of coverage compared with Eflomal).\n[fig:example]\nConcept stability\n-----------------\nWe define the crosslingual semantic field $\\cal F$ of focal concept $F \\in {\\cal S}$ as the second neighborhood of *F*, the set of nodes at a distance 2 from *F*:\n \n$$\\begin{aligned}\n {\\cal F}(F) = \\{ S \\in {\\cal S}| \\exists c: (F,c) \\in {\\cal G} \\wedge\n(c,S) \\in {\\cal G} \\}\\end{aligned}$$ \nshows the crosslingual semantic field of ‘bird’. The strength of the line connecting and *S* (which contains an English string) indicates the number of languages through which can reach *S*., and have thick lines, indicating that there are many languages for which connects to a target string whose meaning includes a bird species. The size of a node indicates the number of paths from to the string the node represents. For example, the size of the node (i.e., *F*) indicates the number of recurrent paths, i.e., $|\\{c|(F,c) \\in {\\cal G} \\wedge (c,F) \\in {\\cal G}\\}|$. The visualization in suggests that is *stable* crosslingually: if we go roundtrip from English to a target language *l* and back, in most cases what we get is. This is often not true (as we will see shortly) for a more abstract concept like. The proportion of recurrent paths is small: many paths starting from go to other nodes, such as and, indicating that it is unstable. See for visualizations of all 83 concepts.\nWe define the *stability* *σ*(*F*) of a focal concept $F \\in \\cal S$ as:\n \n$$\\sigma(F) = \\frac{|\\{ c | (F,c) \\in {\\cal G} \\wedge (c,F) \\in {\\cal G}\\} |}\n{|\\{ c | (F,c) \\in {\\cal G}\\}|}$$ \nThus, for a stable concept *F* (one whose stability is close to 1.0), most paths starting from *F* are part of a “recurrent” path that eventually returns to *F*. In contrast, an unstable concept *F* like has relatively fewer such recurrent paths and a large proportion of its paths go to other concepts.\ncccc Accuracy & Precision & Recall & *F*1 \n0.71 & 0.65 & 0.88 & 0.75 \nWe hypothesize that one cause of stability is concreteness: concrete concepts are more stable across languages than abstract ones because they are directly grounded in a perceptual reality that is shared across languages. To test this hypothesis, we define a concept to be concrete (resp. abstract) if its concreteness score *γ* according to is *γ* ≥ 3.5 (resp. *γ* ≤ 2.5). 69 of our 83 concepts are either abstract or concrete, according to this definition (see Tables [tab:conceptstability1] and [tab:conceptstability2] in the Appendix for concreteness and stability measures of all 83 concepts). We define a concept *F* to be stable iff *σ*(*F*) ≥ 0.6. shows that when we predict stability based on concreteness (i.e., a concept is predicted to be concrete iff it is stable), accuracy is high: *F*1 = .75. This is evidence that our hypothesis is correct: concreteness is an important contributor to stability. See for further analysis of the stability of concepts.\nLanguage similarity\n-------------------\nrc@\n|cccccc|c\n& & & & & & & & \n & 32 &.21 &.2 &.53 &.09 &.14 &.00 &.13 \n& 51 &.24 &.19 &.26 &.08 &.04 &.03 &.11 \n& 83 &.29 &.31 &.49 &.11 &.14 &.04 &.17 \n & 32 &.54 &.41 &.80 &.24 &.39 &.15 &.29 \n& 51 &.52 &.45 &.48 &.18 &.12 &.09 &.24 \n& 83 &.63 &.51 &.77 &.31 &.28 &.09 &.32 \n & 32 &.63 &.49 &.85 &.30 &.43 & &.33 \n& 51 &.64 &.57 &.57 &.20 &.13 &.13 &.30 \n& 83 &.74 & &.83 &.40 &.37 &.12 &.37 \n & 32 &.68 &.53 & **.87** &.34 & & **.18**&.36 \n& 51 &.71 &.59 &.60 &.22 &.14 &.15 &.32 \n& 83 & & & & **.42** &.36 & **.18** &**.39** \n & 32 &.73 &.56 &.84 &.34 & **.54** & **.18** &.37 \n& 51 &.74 & **.61** &.61 &.21 &.09 &.12 &.32 \n& 83 & **.80** & **.61** &.83 & &.28 & &\nWe now propose and evaluate a new measure of similarity between languages, *conceptual similarity*, based on conceptualization. Since we have aligned concepts across languages, we can compute measures of how similar the conceptualization of two languages is. For example, in contrast to Western European languages, Chinese, Korean, and Japanese have one concept that means both and. Our measure aggregates such patterns over many concepts and predicts higher similarity between the three East Asian languages and lower similarity to Western European languages.\nTo compute conceptual similarity, we represent a language *l* as the concatenation of 83 vectors *v⃗*(*l*, *F**j*), each capturing how it represents one of our 83 concepts:\n \n$$\\begin{aligned}\n \\vec{v}(l) = [\\vec{v}(l,F\\_1);\\vec{v}(l,F\\_2);\\ldots;\\vec{v}(l,F\\_{83})]\\end{aligned}$$ \nwhere [; ] is vector concatenation. We define *v⃗*ʹ(*l*, *F**j*) as a 100-dimensional vector and set\n \n$$\\begin{aligned}\n \\vec{v}'(l,F\\_j)\\_i=\n|\\{ c | (F\\_j,c) \\in {\\cal G} \\wedge (c,\\{e\\_i\\}) \\in {\\cal\nG}|\\end{aligned}$$ \ni.e., the number of paths from *F**j* to the English ngram *e**i*; here we only consider nodes *c* = (*l*ʹ, *v*, *T*) for which *l*ʹ = *l*, i.e., only nodes that belong to language *l*. For example, connects with Chinese nodes containing in FP. BP connects these nodes not only to, but also to. Our convention is that the first dimension *v⃗*ʹ(*l*, *F**j*)1 always represents the value of the focal concept *F**j*. To define the other dimensions, we sort all associated English ngrams *e**k* according to the number of languages in which they are associated with *F**j* and select the top 99[2](#fn2); these are then the dimensions 2-100 of *v⃗*ʹ(*l*, *F**j*). We compute the final vector *v⃗*(*l*, *F**j*) by normalizing *v⃗*ʹ(*l*, *F**j*) by ∑*k**v⃗*ʹ(*l*, *F**j*)*k*.\n*v⃗*(*l*, *F**j*) captures which concepts related to *F**j* are clustered in *l* and thereby indicates *l*’s similarity to other languages. For example, for the focal concept, the *v⃗*(*l*, *F**j*) for Chinese, Japanese and Korean are more similar, but they are less similar to *v⃗*(*l*, *F**j*) for Western European languages.\nWe can now define the *conceptual similarity* between two languages *l*1 and *l*2 as the cosine similarity between their vectors:\n \n$$\\begin{aligned}\n \\mbox{c-sim}(l\\_1,l\\_2) = \\cos(\\vec{v}(l\\_1),\\vec{v}(l\\_2))\\end{aligned}$$ \nWe evaluate on Glottolog 4.7. We select the six language families that have more than 50 members in the PBC:,,,, and. We then evaluate conceptual similarity on a binary classification task: *Is the majority of language *l*’s *k* nearest neighbors in the same family as *l*?* In addition to representations based on all 83 focal concepts (referred to as **All83**), we also analogously create representations based just on Swadesh32 and Bible51.\nshows that for two “dense” families (i.e., most members have close relatives), our results are good (up to.8 for,.87 for ). For, and, about half of the predictions are correct for the best *k*. performance is bad, indicating that languages are conceptually more distant from each other. The difference between Swadesh32 and Bible51 performance is large in some cases, especially for and. We hypothesize that the conceptualization for more abstract concepts in Bible51 is more variable than for more concrete concepts in Swadesh32.\nThe main result of this evaluation is that the language representations (i.e., the *v⃗*(*l*)) derived by are a good basis for assessing the similarity of languages. discusses in detail the complementarity of conceptual similarity to other similarity measures.\n*stability* measure vs. *concreteness* scores.\nAnalysis\n========\nConcept stability\n-----------------\nFigure [fig:stabilityconcreteness] and Table [tab:concretenessstabilitybinary] support our hypothesis that concreteness contributes to stability, as most concrete concepts tend to be stable and most abstract concepts tend to be less stable. However, recall is higher than precision in Table [tab:concretenessstabilitybinary], indicating that stable concepts are mostly concrete but not vice versa, as plenty of concepts are located in the lower right in Figure [fig:stabilityconcreteness]. In our analysis, we find that some concrete concepts are unstable because meaning is extended by semantic processes like metonymy and metaphor. We now give examples of our analysis of this phenomenon in. See for the visual semantic fields of all 83 concepts.\nshows (i) concrete concepts whose instability is plausibly caused by metaphor, metonymy, and other semantic extensions and (ii) crosslingually associated English concepts that indicate instability. For example, metonymy, where a concept is represented by the name of something associated with it, results in instability frequently. This is in line with crosslinguistic studies that show that metonymy seems to be a universal language phenomenon.\nlll types & concept & associated \n&‘ear’ & *hear*, *listen* \n& ‘mouth’ & *word*, *speak* \n & ‘sun’ & *day*, *east* \n& ‘moon’ & *month* \n& ‘belly’ & *womb*, *birth* \n& ‘soldier’ & *army*, *military* \n & ‘seed’ & *son*, *offspring* \n& ‘mouth’ & *entrance* \n & ‘water’ & *waters*, *sea*, *drink* \n& ‘knee’ & *obeisance*, *worship* \nProcesses of semantic extension are central to cognitive linguistics for explaining historical changes, language use and framing in discourse. Since these processes are pervasive in communication, in the lexicon and in historical language change, they are also a plausible cause for crosslingual instability of concepts.\nc\n[fig:clustering]\nLanguage similarity\n-------------------\nshows a visualization of conceptual similarity using t-SNE. Each point in the figure represents a language. Color indicates family on the left (for the six families from ) and area on the right (for all languages, area extracted from Glottolog 4.7).\nThe right figure (area) shows that conceptual similarity is correlated with geographic proximity: except for, all regions are concentrated in a few subclusters, with being most clustered and least. For example, there are two clusters (top left, top center) that cover most languages. This is consistent with findings that close languages often influence each other. Of course, the main reason is the correlation between conceptual and typological relatedness (see below) and between typological and geographic proximities. But there are also indications of geography-induced conceptual similarity, e.g., most languages have close neighbors. We present examples at the end of.\nThat geographic proximity is not sufficient for conceptual similarity is demonstrated by : it is distributed over most of the figure, indicating that the underlying conceptualizations are quite different. This is consistent with the well-known fact that exhibits enormous linguistic diversity in a very small geographic area. Of course, there is still subcluster structure: most languages have near neighbors in conceptual space.\nTurning to the left figure (family), we see that there is agreement of conceptual similarity with typological similarity. Some families form a relatively clear cluster, in particular,, suggesting that languages are similar in conceptualization. This explains ’s high accuracy in.\nBut there is also complementarity between conceptual similarity and typological similarity. is spread out across the entire figure, explaining its low accuracy in. For Chinese and Tibetan, we find their conceptualizations to be quite different, in particular, for body parts (e.g., `mouth` and `neck`). See for examples.\nConversely, we now present three examples of typologically distant languages that are still conceptually close. The first example is **Tagalog**. While languages mostly occur in a relatively tight region of the figure, that region also contains many non-Indo-European languages. We hypothesize that languages have influenced the conceptualization of other languages worldwide due to their widespread use, partly as part of colonization. One example is that the Tagalog words and mean both and, a conceptualization similar to Spanish (), a language Tagalog was in close contact with for centuries. Standard Malay is typologically related to Tagalog, but its word for, does not include the meaning. This may contribute to Tagalog being conceptually more similar to Spanish on our measure than other languages. **Plateau Malagasy**, an language spoken in Madagascar, is conceptually similar to both far-away languages like Hawaiian (reflecting its typology) as well as to geographically close, but typologically dissimilar languages like Mwani and Koti. **Masana** is an Afro-Asiatic language spoken in Nigeria. It is conceptually close to the languages Yoruba, Igbo and Twi, also spoken in and around Nigeria. Geographic proximity seems to boost conceptual similarity in these three cases. We leave further investigation of the hypothesis that -based representations reveal historical interlanguage influences to future work.\nConclusion & Future Work\n========================\nWe propose, a method that automatically aligns source-language concepts and target-language strings by creating a directed bipartite graph. We investigate the structure of such alignments for 83 focal concepts. Our extensive manual evaluation demonstrates good performance of. We introduce the notion of crosslingual stability of a concept and show, using, that concrete concepts are more stable across languages than abstract concepts. We also define conceptual similarity, a new measure of language similarity based on representations. In our experiments, conceptual similarity gives results that partially agree with established measures like typological and areal similarity, but are complementary in that they isolate a single clearly defined dimension of similarity: the degree to which the conceptualization of two languages is similar.\nIn the future, we would like to improve the efficiency of and, extending our work on a sample of 83 in this paper, apply it to all concepts that occur in PBC.\nLimitations\n===========\nThe we propose consists of two core steps, i.e., forward pass and backward pass. The forward pass identifies the most associated target-language strings for a focal concept. However, due to possible data sparsity of PBC in some low-resource languages and some cases of verse-level misalignment, *χ*2 scores of the real translations can be indistinguishable compared with some other rare words that also occur in the same verses. Under such rare cases, will not work well enough. In addition, the genre of PBC is limited to religion and therefore the diversity of the concepts across languages is largely influenced. Nevertheless, PBC, as far as we know, provides texts in the largest number of low-resource languages. PBC is thus a good fit for our goal.\nIn this work, we select 83 concepts, including the Swadesh32 and Bible51, representing a wide range of interesting crosslingual concepts. The runtime for computing the results for one concept in all languages is around 10 hours on average. The relatively long runtime, however, can prevent us from exploring more interesting concepts.\nWe find that the concreteness of a focal concept can be a contributor to the stability measure. As we use English as the source language for representing the focal concepts, we naturally resort to concreteness scores from English language ratings only. In addition, the analysis is carried out from an English perspective. Nevertheless, as we want to compare different languages, we have to use a unified source language. Theoretically, we can use any language as the source language and represent the concepts in that language. We therefore plan to use other languages, e.g., Chinese, or some low-resource languages, as the source language in future research.\nEthics Statement & Risks\n========================\nIn this work, we investigate the differences in conceptualization across 1,335 languages by aligning concepts in a parallel corpus. To this end, we propose, a method that creates a directed bipartite alignment graph between source language concepts and sets of target language strings. The corpus we used, i.e., PBC, contains translations of the Bible in different languages (one language can have multiple editions). As far as we know, the corpus does not include any information that can be used to attribute to specific individuals. Therefore, we do not foresee any risks or potential ethical problems.\nAcknowledgments\n===============\nWe would like to acknowledge Verena Blaschke for her valuable suggestions. We would also like to thank the reviewers for their positive and constructive feedback. This work was funded by the European Research Council (grant #740516).\nDetails of data\n===============\nParallel Bible Corpus\n---------------------\nWe work with the Parallel Bible Corpus (PBC, ), which contains 1,775 editions of the Bible in 1,335 unique languages (we regard dialects with their own ISO 639-3 codes[3](#fn3) as different languages). As far as we know, there is no explicit licence for PBC dataset. For each language, we only use one of its available editions. We use the *New World* edition for each language, if available, and the edition with the largest number of verses otherwise. Different from previous work which only used verses that are available in all languages, we use all parallel verses between English and any other target languages. This means the number of parallel verses between English and other languages can be different. In general, we have parallel verses with a number greater than 30,000 (Hebrew + New Testament) for high-resource languages (141 languages), e.g. French, German and Chinese, while around 7,900 (New Testament only) for most of the other languages (1038 languages).\nConcept selection\n-----------------\nWe have 83 focal concepts in total which are listed in Tables [tab:conceptstability1] and [tab:conceptstability2]. We classify the focal concepts into Swadesh32 and Bible51, for which we explain the selection of concepts in detail as follows.\nThe *Swadesh 100* list offers 100 English words that represent universal and basic concepts. The words include nouns, adjectives, and verbs. We limit our selection to nouns to facilitate the comparison between concepts. Because we choose Bible editions as our resource, many concepts in the list can have very low frequencies of occurrences or even do not occur at all. On the contrary, some concepts in the list can occur many times but are less interesting to us, e.g., ‘I’, ‘you’, ‘we’, ‘this’ etc. We therefore only keep the concepts in the list which occur equal to or more than 5 times in the New Testament (only New Testament is available for most low-resource languages so the concept has to appear in the New Testament) and less than or equal to 500 times in Hebrew + New Testaments. There are 32 concepts that fulfill the criterion and we refer them to as **Swadesh32**, which are shown in Table [tab:conceptstability1].\nFor Bible concepts, we first obtain the distinct types of strings that have a length between 4 and 15 characters from all words in the English Bible. The ngrams can contain but cannot go beyond the word boundaries. For example, from a part of sentence `$bird$fly$` (substituting whitespaces with `$`), we can obtain ngrams such as `$bird`, `$bird$` and `fly$`, but `$bird$f` is not possible because it contains `$` in the middle. After this, we randomly select 10 languages from the available languages of PBC plus Chinese and German (12 languages in total). For each of the selected languages, we compute the coverage of the identified most associated target-language string obtained by performing a forward pass (setting the max number of iterations *M* to 1) (Algorithm [alg:fs]) by regarding each English string as a focal concept. If the coverage of a string is larger than 0.5 for more than five languages, then we keep it otherwise we filter it out. This results in around 1000 strings. Then we filter out those that represent named entities. Finally, we manually check the list and select the strings that represent nouns and are not in the Swadesh list and are more or less specific to the Bible. This finally results in a set of 51 concepts (**Bible51**) which are shown in Table [tab:conceptstability2].\nAdditional details of\n=====================\nFocal concepts & strings\n------------------------\nThe bipartite graph $\\cal G$ we construct contains source nodes set $\\cal S$ and target nodes set $\\cal T$. Each node $s \\in \\cal{S}$ is a concept and is represented by a set of strings. We restrict the length of the strings between 1 and 8 for any language except the source language English (we restrict the length larger than 2 and the strings cannot go beyond word boundaries) for efficiency. To differentiate the nodes in ${\\cal S}$, we refer to the set of our chosen 83 focal concepts as ${\\cal S}\\_F \\subset {\\cal S}$. Each focal concept *F* in ${\\cal S}\\_F$ is a set which can contain multiple strings, e.g., ‘belly’ concept: {`\\$belly\\$`, `\\$bellies\\$`}. In contrast, other concepts in ${\\cal S}$ are sets which contain only a single string, e.g., {`\\$sparrows\\$`}. In the backward pass for a focal concept *F*, if a string *s* being identified belongs to *F*, then we create an edge that ends at *F* instead of *s*.\nString candidates\n-----------------\nThe consists of (1) a forward pass: for building edges $(F, c) \\in {\\cal S} \\times \\cal{T}$ and (2) a backward pass for building edges $(c, f) \\in {\\cal T} \\times {\\cal S}$. In the forward pass, for example, an edge $(F, c) \\in {\\cal S} \\times {\\cal T}$ is constructed if a target node from the target language *l*: *c* = < *l*, *v*, *T* > ’s verse *v* contains a string *t* that is highly associated with *F*. As the search space of target-language strings is extremely large for each *l*, we therefore restrict the search space to the set of strings which occur in the verses whose corresponding English verses contain *F*. Formally, let Π(eng, *F*) be the verses where the focal concept *F* occurs and ${\\cal P}(\\Sigma^\\*)(l, v)$ be the strings that fulfill our string selection conditions in verse *v* for language *l*. We will then only consider the strings in $\\mathbb{T} = \n\\bigcup\\_v {\\cal P}(\\Sigma^\\*)(l, v) | v \\in \\Pi(\\text{eng}, F)$ to be candidates in language *l* that are possibly associated with *F*. Similarly, in the backward pass, we also restrict the search space to be the set of English strings in the verses whose corresponding target-language verses contained the identified target-language string set *T* in the forward pass. Formally, let Π(*l*, *T*) be the verses where the *T* occurs and ${\\cal P}(\\Sigma^\\*)(\\text{eng}, v)$ be the strings that fulfill our string selection conditions in verse *v* for English. We will then only consider English strings in $\\mathbb{S} = \n\\bigcup\\_v {\\cal P}(\\Sigma^\\*)(\\text{eng}, v) | v \\in \\Pi(l, T)$. Furthermore, we will only consider the strings *t* ∈ T which occur more than Π(eng, *F*)/10 in the target-language verses of Π(eng, *F*) and *s* ∈ S which occur more than 2 times in the English verses of Π(*l*, *T*).\nMeasuring association\n---------------------\nGiven a set of strings *U* in a language *l*2, we want to find a string in language *l*1 that is associated with *U*. To this end, we use *χ*2 score to measure the degree of the association. Specifically, we divide the verse set into two subsets: verses containing *U* and verses not containing *U* in the Bible of *l*2, i.e., Π(*l*2, *U*) and ¬Π(*l*2, *U*). We then build a contingency table for each string candidate $t \\in {\\cal P}(\\Sigma^\\*)$ and *t* comes from language *l*1, as shown in Table [tab:cont]. After that, we compute the *χ*2 score for each string: the higher the score, the more associated the string in *l*1 is with the set of strings *U*. We then choose the string that has the highest *χ*2 score as a hit in language *l*1 for a set of strings *U* in language *l*2.\n| | in Π(*l*2, *U*) | in ¬Π(*l*2, *U*) |\n| --- | --- | --- |\n| *I*(*t*) | *n*00 | *n*01 |\n| ¬*I*(*t*) | *n*10 | *n*11 |\nAdding edges\n------------\nFor efficiency and stability reasons, in the actual implementation of the backward pass, does not add edges from a target node to all the strings in *S* that fulfill the criterion, i.e., the set of the identified associated source language strings, as shown in Algorithm [alg:bsnew] (line 9). Instead, we only add one edge only from each target node. This means that in each iteration of the backward pass, we will add new edges starting from the involved target nodes to a single *s*: $\\{((l,v,T),s)|(F, (l,v,T)) \\in {\\cal G} \\wedge s \\in v\\ \\wedge v \\in V\\}$. In this way, each target node that was previously connected with the focal concept *F* can only be connected to one source node only. By doing this, we find that some undesirable associations can be avoided.\nHyperparameters\n---------------\nWe have two hyperparameters in, i.e., (1) the maximum number of iterations of searching associated strings for a focal concept in each language: *M* and (2) the threshold *α* for the minimum coverage of the set of identified associated string *U*. We set *M* = 5 and *α* = .9 as default values for all involved computations. Based on preliminary experiments on ‘bird’ concept, we found that the number of associated strings usually will not go beyond 5. Moreover, when *M* is large, we might have an efficiency problem (more iterations for each language) when computing other focal concepts. Therefore, we set *M* = 5 to reduce the runtime of while not sacrificing the accuracy too much. As for the coverage threshold *α*, we conduct preliminary experiments with.85,.9 and.95 respectively on ‘bird’ concept. We found when coverage is small (<.9), the search stops when there are still possibly unidentified associated strings in the rest verses. If the coverage is too large (>.9), we found that some less related strings can be identified at the later stage of iterations for some languages. We should note that PBC can have verse-level misalignment problems, which means for some parallel verses ‘bird’ occurs in English but the target-language verse can be unrelated to ‘bird’ at all. Moreover, as we remove the parallel verses that we have covered in each iteration, the verses uncovered become smaller and smaller in each iteration. *χ*2 scores computed on later iterations can be not significant and multiple strings can have the same highest *χ*2 scores if they only appear in the uncovered verses with the same number of occurrences. Therefore, to ensure that finds enough strings while guaranteeing the quality of the associations between them with the search string, we set the coverage threshold *α* = .9.\n![Part of the document for annotation for one target language, i.e., aai (Arifama-Miniafia) The true positive and false positive verses are not shown here.](./figs/annotation_example.jpg \"fig:\")[fig:annotation]\n[fig:categories]\nDetails of Evaluation\n=====================\nSingle concept across all languages\n-----------------------------------\nIn the English version of the Bible, we find the ‘bird’ concept is often expressed by the following words/phrases: “bird(s)”, “fowl(s)”, “flying creature(s)” and “winged creature(s)”. Therefore, we use the following strings: `$bird`, `$fowl`, `$flying$creature` and `$winged$creature` to represent the ‘bird’ concept. We agree that this set of strings might not be optimal for other large-resource parallel datasets (if there are any). However, for PBC dataset, this set of strings could empirically cover the ‘bird’ concept. We perform our which includes the forward pass and backward pass for this ‘bird’ concept for all the languages in PBC. We perform three types of evaluations as follows based on the FP and BP hits:\n### ‘bird’ conceptualization in all languages\nWe provide a pdf document in which the statistics of each string identified in both forward pass and backward pass are shown. In addition, for each target-language string, a) two randomly sampled **True Positive** parallel verses, i.e., target-language verses that contain the identified strings and the parallel English verses contain ‘bird’; b) two randomly sampled **False Positive** parallel verses, i.e., target-language verses that contain the strings and the parallel English verses that do not contain ‘bird’, are shown. We also show three randomly sampled **False Negative** parallel verses, i.e., target-language verses that do not contain the strings and the parallel English verses that contain ‘bird’ concept. An example of part of the document for one language is shown in Figure [fig:annotation]. By checking the general patterns demonstrated in the document, we define four evaluation categories: **one-to-one**, **polysemy**, **ambiguity** and **failure**. Noticeably, our category **polysemy** and **ambiguity** do not directly correspond to the definition in linguistics, but reflect general patterns of the conceptualization. The classification of these two categories is based on the pattern of strings identified in the backward pass. More specifically, we classify the conceptualization pattern of a language as **polysemy**, if it shows one of the following patterns: (a) *hyponymy*, where we found strings such as `dove` and `sparrows`, which are hyponyms of ‘bird’. (b) *meronymy*, where we found strings like `$wings`, which is a meronym of ‘bird’. (c) *other related words*, where we found strings such as `$fly` and `$chirp`, which are apparently related to ‘bird’, but do not fit into any well-defined lexical semantic relation. The conceptualization pattern is classified as **ambiguity**, if the strings we found in the backward pass are not semantically related to ‘bird’ at all (such as `$new$` and `$kid$`), but nevertheless are deemed as highly associated with ‘bird’ by our algorithm. These cases are generally caused by having homonyms of ‘bird’ in the target language. In case the linguist annotator cannot be sure of the classification, consultation has been made with other experts to resolve these issues and find the most common agreement.\n| *l* | *P* | *T* | category |\n| --- | --- | --- | --- |\n| ify | ke:keq, qemayuq | `$sisit$` | **no overlap** |\n| ind | cewek, burung | `$burung`, `terbang$` | **overlap** |\n| akb | unggas | `$unggas$` | **match** |\n| afr | voël, vliegtuig | `voël` | **match** |\n| *l* | *P* | *T* | factor |\n| --- | --- | --- | --- |\n| uig | qush | (qush) | **script** |\n| mua | | `juu` | **transcription** |\n| lip | | | **prefix** |\n| mse | layra | `layagi` | **suffix** |\n| sbl | ma`nok | `manokmanok` | **reduplication** |\n### Translations compared with PanLex\nFor each language, the linguist annotator also checks the translation of “bird” provided by PanLex [4](#fn4). The translations are available in 1,137 languages out of the 1,331 languages in PBC where we found translations of the focal concept. We define the following four categories for the PanLex evaluation where *P* are the tranlations from PanLex and *T* are the FP hits (target-language strings).\n* **no translation**: *P* = ∅, i.e., PanLex gives no translation.\n* **no overlap**: *T* ∩ *P* = ∅, i.e., none of the FP hits is found in PanLex translations.\n* **overlap**: 0 < ∣*T* ∩ *P*∣ < ∣*T*∣, i.e., some but not all of the FP hits are found in PanLex translations.\n* **match**: ∣*T* ∩ *P*∣ = ∣*T*∣, i.e., all the FP hits can be found in PanLex translations. Note that we do not require all the translations in PanLex to be present in our set of target strings, since PanLex often gives a very long list of translations and our goal is to use PanLex translations to confirm the strings we identified.\nWe show examples for each category (except for **no translation**) in Table [tab:panlexexamples]. When deciding whether a translation from PanLex matches an FP hit, the linguist annotator does not only look for an exact match of strings but also takes the differences in scripts, transcriptions, and morphological forms into consideration. For languages with multiple writing systems, words are naturally transliterated into a unified script for them to be comparable. It has also been observed, that the same word can sometimes be transcribed differently in different sources, especially for low-resource languages that have no standard writing system. Therefore, “” in PanLex and “juu” in our FP hit will still be considered as a match. Furthermore, it is also possible that the PanLex translation uses a morphologically different form compared to our FP hit, such as dictionary form versus inflected form. Possible morphological processes such as affixation, reduplication, vowel mutation, vowel ellipsis, and metathesis have been taken into consideration. We show some examples of these factors in Table [tab:panlexnotexactmatch].\nWith careful examination, if the linguist annotator concludes that actually has found the same lexeme as the PanLex translation, and the difference between PanLex and our FP hit is merely attributed to the above-mentioned factors, they will still be considered as matches.\n| model | Coverage(g) | Coverage(a) | trans. per *l* |\n| --- | --- | --- | --- |\n| |.93 |.95 | 1.6 |\n| Eflomal 0 |.94 |.96 | 7.0 |\n| Eflomal 1 |.81 |.82 | 2.9 |\n| Eflomal 0.1 |.70 |.79 | 2.1 |\n### Translations compared with Eflomal\nWe also compare the coverage and the average number of translations proposed per language of and eflomal, a statistical word alignment tool, in Table [tab:birdresultsvseflomal]. We collect words that are aligned with one of the strings representing ‘bird’ in all verses where ‘bird’ occurs for eflomal baselines. The coverage means the fraction of verses containing ‘bird’ covered by the set of proposed translations. Global coverage denotes that we compute the coverage directly in all verses regardless of language while average coverage denotes that we first compute the coverage for each language and then average over all languages. We notice that, eflomal, without filtering any translations, obtains the highest global and average coverage, which can be regarded as the upper bound. However, the number of translations per language on average is so high: 7.0. After filtering some proposed translations by their frequencies (1 and 0.1), we observe a sudden drop in the coverage. This indicates that (1) eflomal can propose many wrong alignments and (2) some correct alignments have very small frequencies. Because of its word-level alignment nature, eflomal cannot take the possible morphological changes of the words into consideration at all. On the contrary, only proposes 1.6 translations on average while keeping the coverage very close to the upper bound, suggesting that can identify the strings (ngrams) that are most associated with the concept and alleviate the possible problems caused by, e.g., morphological changes, in many languages.\nSwadesh concepts\n----------------\nWe resort to NoRaRe [5](#fn5) to find the available translations of the 32 Swadesh concepts in 39 languages (NoRaRe covers). For each concept and each language, we store a triple indexed by the concept-language pair: \n < concept, language, translation(s) > \n We finally obtain 582 triples for evaluation (NoRaRe does not provide translations of a concept in all covered languages). We use *T* to represent the set of translations proposed by and *N* for NoRaRe translations (as ground-truth translations) in the triple. When judging whether a translation in *N* matches a translation in *T* generated by, we do the match leniently to allow for morphological changes. Specifically, if a translation in *N* is a substring of a translation in *T* generated by or the other way around, we regard it as a successful match. This is because *N* often provide the dictionary forms of the nouns but *T* are generated automatically based on the actual Bible verses where the nouns can change their suffixes or prefixes quite often depending on their roles in the verses.\nWe are especially interested in the triples in which our identified strings *T* do not match the ground-truth translations *R*, i.e., *T* ∩ *R* = ∅. We sampled 10 such triples (we provide *N* and *T* for each triple) in Table [tab:swadesherrors]. We notice that there are cases where the ground-truth translations *N* use different versions of transliterations. For example, `$andjing` vs. “anjing”, `$daoen` vs. “daun” and `$boelan` vs. “bulan” in Malay (msa). Moreover, there can be multiple equivalent translations for a concept, but *N* just lists one of them which is not used (or not identified) in PBC, e.g., ``\nUTF8gbsn颈\n” is a simpler but more formal translation of ‘neck’ but the *N* only lists ``\nUTF8gbsn脖子\n” in Chinese (zho); the concept ‘path’ can also be translated to “rout” and “sentier” in French (fra) but only “chemin” is given by the *N*. Therefore, we see that this evaluation compared with NoRaRe can actually underestimate the performance of our method.\nllll concept & *l* & *N* & *T* \n‘dog’ & msa & anjing & `$andjing` \n‘seed’ & cym & hedyn & `$had$`, `$heu` \n‘leaf’ & msa & daun & `$daoen` \n‘horn’ & est & ruupor & `sarve` \n‘mouth’ & tur & ağiz & `$ağzı`, `$ağız` \n‘neck’ & zho &\nUTF8gbsn脖子\n&\nUTF8gbsn颈\n \n‘moon’ & msa & bulan & `$boelan` \n‘water’ & cym & dŵr & `dwfr$`, `dyfr` \n‘rain’ & msa & hujan & `$hoedjan` \n‘path’ & fra & chemin & `$la$rout`, `$sentier` \n| Concepts | Concreteness | Stability |\n| --- | --- | --- |\n| ‘fish’ | 5.0 | 0.86 |\n| ‘bird’ | 5.0 | 0.68 |\n| ‘dog’ | 4.85 | 0.85 |\n| ‘tree’ | 5.0 | 0.64 |\n| ‘seed’ | 4.71 | 0.38 |\n| ‘leaf’ | 5.0 | 0.74 |\n| ‘root’ | 4.34 | 0.78 |\n| ‘flesh’ | 4.59 | 0.36 |\n| ‘blood’ | 4.86 | 0.69 |\n| ‘horn’ | 5.0 | 0.82 |\n| ‘hair’ | 4.97 | 0.77 |\n| ‘ear’ | 5.0 | 0.46 |\n| ‘mouth’ | 4.74 | 0.49 |\n| ‘tooth’ | 4.89 | 0.91 |\n| ‘tongue’ | 4.93 | 0.61 |\n| ‘foot’ | 4.9 | 0.7 |\n| ‘knee’ | 5.0 | 0.38 |\n| ‘belly’ | 4.8 | 0.4 |\n| ‘neck’ | 5.0 | 0.72 |\n| ‘breast’ | 4.89 | 0.65 |\n| ‘sun’ | 4.83 | 0.49 |\n| ‘moon’ | 4.9 | 0.48 |\n| ‘star’ | 4.69 | 0.87 |\n| ‘water’ | 5.0 | 0.48 |\n| ‘rain’ | 4.97 | 0.68 |\n| ‘stone’ | 4.72 | 0.71 |\n| ‘cloud’ | 4.54 | 0.68 |\n| ‘smoke’ | 4.96 | 0.57 |\n| ‘path’ | 4.41 | 0.35 |\n| ‘mountain’ | 4.96 | 0.64 |\n| ‘white’ | 3.89 | 0.77 |\n| ‘night’ | 4.52 | 0.68 |\n| Concepts | Concreteness | Stability |\n| --- | --- | --- |\n| ‘babe’ | 3.67 | 0.59 |\n| ‘hypocrit’ | 2.43 | 0.81 |\n| ‘soldier’ | 4.72 | 0.49 |\n| ‘scroll’ | 4.11 | 0.57 |\n| ‘demon’ | 3.32 | 0.45 |\n| ‘boat’ | 4.93 | 0.71 |\n| ‘olive’ | 4.9 | 0.8 |\n| ‘prayer’ | 3.28 | 0.32 |\n| ‘mercy’ | 1.57 | 0.29 |\n| ‘trumpet’ | 4.86 | 0.83 |\n| ‘angel’ | 3.82 | 0.88 |\n| ‘prison’ | 4.68 | 0.62 |\n| ‘savior’ | 3.04 | 0.49 |\n| ‘tomb’ | 4.73 | 0.61 |\n| ‘husband’ | 4.11 | 0.47 |\n| ‘bride’ | 4.63 | 0.69 |\n| ‘talent’ | 2.19 | 0.83 |\n| ‘peace’ | 1.62 | 0.72 |\n| ‘secret’ | 2.19 | 0.57 |\n| ‘faith’ | 1.63 | 0.59 |\n| ‘woe’ | 1.96 | 0.8 |\n| ‘throne’ | 4.64 | 0.62 |\n| ‘wisdom’ | 1.53 | 0.54 |\n| ‘disciple’ | 3.29 | 0.73 |\n| ‘obeisance’ | NA | 0.37 |\n| ‘truth’ | 1.96 | 0.4 |\n| ‘memor’ | 2.83 | 0.53 |\n| ‘governor’ | 4.07 | 0.52 |\n| ‘poor’ | 2.7 | 0.63 |\n| ‘blind’ | 4.03 | 0.77 |\n| ‘spiritual’ | 1.79 | 0.33 |\n| ‘justice’ | 1.45 | 0.34 |\n| ‘courage’ | 1.52 | 0.53 |\n| ‘purpose’ | 1.52 | 0.3 |\n| ‘generation’ | 1.96 | 0.56 |\n| ‘contrary’ | 1.56 | 0.46 |\n| ‘prophesy’ | 2.11 | 0.41 |\n| ‘decision’ | 2.19 | 0.36 |\n| ‘request’ | 2.59 | 0.32 |\n| ‘weakness’ | 2.59 | 0.55 |\n| ‘journey’ | 2.57 | 0.39 |\n| ‘public’ | 2.57 | 0.23 |\n| ‘appearance’ | 2.57 | 0.55 |\n| ‘expression’ | 2.54 | 0.51 |\n| ‘marriage’ | 2.51 | 0.51 |\n| ‘wrath’ | 2.42 | 0.4 |\n| ‘trouble’ | 2.25 | 0.45 |\n| ‘promise’ | 2.09 | 0.46 |\n| ‘power’ | 2.04 | 0.41 |\n| ‘pleasure’ | 2.04 | 0.35 |\n| ‘thought’ | 1.97 | 0.39 |\nInfrastructure & environment\n============================\nWe ran all our computational experiments on a CPU server with 48 cores and 1024 GB of memory. We used Python 3.6[6](#fn6) throughout our implementation of and for visualizations. Specifically, for fundamental scientific computing (e.g., computing *χ*2 scores), we used NumPy[7](#fn7), SciPy[8](#fn8) and scikit-learn[9](#fn9) packages. For visualization, we used NetworkX[10](#fn10)(mainly for the crosslingual semantic fields) and Matplotlib[11](#fn11) packages.\nCrosslingual semantic fields\n============================\n[fig:semanticfield1]\n[fig:semanticfield2]\n[fig:semanticfield3]\n[fig:semanticfield4]\n[fig:semanticfield5]\nFurther analysis regarding language similarity\n==============================================\n[htbp] [fig:clustering5183]\nlcccccccc setting & global & atla & aust & indo & guin & otom & sino \nSwadesh32 & 1280 & 222 & 215 & 91 & 87 & 76 & 68 \nBible51 & 1264 & 219 & 210 & 90 & 85 & 76 & 67 \nAll83 & 1263 & 219 & 210 & 90 & 85 & 76 & 67 \nlcccccccc #neighbors & global & Papunesia & Africa & Eurasia & North America & South America & Australia \nk=1 & 0.51 & 0.51 & 0.52 & 0.64 & 0.52 & 0.35 & 0.00 \nk=2 & 0.28 & 0.27 & 0.31 & 0.47 & 0.27 & 0.09 & 0.00 \nk=3 & 0.45 & 0.44 & 0.51 & 0.62 & 0.43 & 0.22 & 0.00 \nk=4 & 0.53 & 0.52 & 0.60 & 0.69 & 0.51 & 0.25 & 0.00 \nk=5 & 0.55 & 0.54 & 0.61 & 0.72 & 0.57 & 0.28 & 0.09 \nk=6 & 0.56 & 0.55 & 0.62 & 0.75 & 0.57 & 0.26 & 0.09 \nk=7 & 0.56 & 0.55 & 0.63 & 0.74 & 0.56 & 0.24 & 0.09 \nk=8 & 0.57 & 0.56 & 0.64 & 0.76 & 0.56 & 0.27 & 0.00 \nk=9 & 0.57 & 0.54 & 0.66 & 0.77 & 0.59 & 0.26 & 0.00 \nk=10 & 0.58 & 0.55 & 0.65 & 0.79 & 0.62 & 0.22 & 0.00 \nlcccccccc #neighbors & global & Papunesia & Africa & Eurasia & North America & South America & Australia \nk=1 & 0.49 & 0.54 & 0.57 & 0.63 & 0.28 & 0.30 & 0.00 \nk=2 & 0.24 & 0.25 & 0.33 & 0.42 & 0.06 & 0.08 & 0.00 \nk=3 & 0.41 & 0.43 & 0.59 & 0.62 & 0.13 & 0.09 & 0.00 \nk=4 & 0.48 & 0.52 & 0.66 & 0.70 & 0.20 & 0.14 & 0.00 \nk=5 & 0.51 & 0.55 & 0.70 & 0.72 & 0.18 & 0.18 & 0.00 \nk=6 & 0.50 & 0.53 & 0.72 & 0.70 & 0.18 & 0.15 & 0.00 \nk=7 & 0.50 & 0.53 & 0.71 & 0.70 & 0.18 & 0.16 & 0.00 \nk=8 & 0.49 & 0.51 & 0.71 & 0.69 & 0.14 & 0.15 & 0.00 \nk=9 & 0.48 & 0.48 & 0.73 & 0.71 & 0.12 & 0.12 & 0.00 \nk=10 & 0.49 & 0.50 & 0.74 & 0.71 & 0.14 & 0.12 & 0.00 \nlcccccccc #neighbors & global & Papunesia & Africa & Eurasia & North America & South America & Australia \nk=1 & 0.54 & 0.59 & 0.65 & 0.68 & 0.38 & 0.25 & 0.00 \nk=2 & 0.33 & 0.35 & 0.41 & 0.53 & 0.16 & 0.09 & 0.00 \nk=3 & 0.51 & 0.51 & 0.69 & 0.71 & 0.27 & 0.17 & 0.00 \nk=4 & 0.56 & 0.57 & 0.75 & 0.75 & 0.32 & 0.21 & 0.00 \nk=5 & 0.58 & 0.59 & 0.77 & 0.78 & 0.34 & 0.22 & 0.00 \nk=6 & 0.58 & 0.60 & 0.77 & 0.79 & 0.35 & 0.17 & 0.00 \nk=7 & 0.58 & 0.60 & 0.79 & 0.81 & 0.34 & 0.15 & 0.00 \nk=8 & 0.57 & 0.59 & 0.77 & 0.81 & 0.33 & 0.16 & 0.00 \nk=9 & 0.58 & 0.59 & 0.78 & 0.81 & 0.32 & 0.18 & 0.00 \nk=10 & 0.58 & 0.60 & 0.79 & 0.80 & 0.32 & 0.17 & 0.00 \n| | | |\n| --- | --- | --- |\n| | zho | `$mouth$`, `$mouths$`, `$entrance`, `alat`, `$ford` |\n| | bod | `$mouth$`, `$mouths$`, `$prais` |\n| [1ex] | zho | `$neck$`, `$necks$`, `$stiff-necked` |\n| | bod | `$neck$`, `$necks$`, `$obstina`, `$stubbornness$` |\n| [1ex] | zho | `$tree$`, `$trees$`, `$vine$`, `-tree$`, `$boughs$` |\n| | bod | `$tree$`, `$trees$`, `$chariot`, `wood`, `$cedar`, `igs$`, `$timber` |\n| [1ex] | zho | `$horn$`, `$horns$`, `$corner` |\n| | bod | `$horn$`, `$horns$`, `$trumpet`, `$fathering$` |\n| | | |\n| --- | --- | --- |\n| | arb | `$fish$`, `$fishes$` |\n| | pes | `$fishes$`, `$fish$`, `$fish` |\n| | tur | `$fish$`, `$fishes$`, `$fish`, `$honey` |\n| | msa | `$fishes$`, `$fish$` |\n| | ind | `$fish$`, `$fishes$` |\n| [1ex] | arb | `$stars$`, `$star$` |\n| | pes | `$stars$`, `$star$` |\n| | tur | `$stars$`, `$star$` |\n| | msa | `$stars$`, `$star$` |\n| | ind | `$stars$`, `$star$` |\n| [1ex] | arb | `$blood$`, `$bloodguilt$` |\n| | pes | `$blood$` |\n| | tur | `$blood$`, `$bloods`, `$bloodguilt$` |\n| | msa | `$blood$` |\n| | ind | `$blood$`, `$blood` |\n| [1ex] | arb | `$tongue$`, `$tongues$` |\n| | pes | `$tongue$`, `$tongues$` |\n| | tur | `$tongue$`, `$tongues$`, `$request`, `$language` |\n| | msa | `$tongues$`, `$tongue$`, `ebrew$` |\n| | ind | `$tongue$`, `$tongues$`, `$language` |\n| [1ex] | arb | `$birds$`, `$bird$`, `$flying$`, `$fowls$` |\n| | pes | `$birds$`, `$bird$` |\n| | tur | `$birds$`, `$bird$`, `$fowl` |\n| | msa | `$birds$`, `$bird$`, `dove`, `$sparrows$`, `$eagle` |\n| | ind | `$birds$`, `$bird$`, `$eagle`, `$turtledove`, `$ostrich`, `$raven`, `$bird` |\n| | | |\n| --- | --- | --- |\n| | eng | `$blood$` |\n| | spa | `$blood$` |\n| | ell | `$blood$` |\n| | rus | `$blood$`, `$blood` |\n| | tgl | `$blood$` |\n| | swh | `$blood$`, `$blood` |\n| | hye | `$blood$`, `$blood` |\n| [1ex] | eng | `$tongue$`, `$tongues$` |\n| | spa | `$tongue$`, `$tongues$`, `$language` |\n| | ell | `$tongue$`, `$tongues$`, `$language` |\n| | rus | `$tongue$`, `$tongues$`, `$language` |\n| | tgl | `$tongue$`, `$tongues$`, `$language` |\n| | swh | `$tongue$`, `$tongues$`, `$language` |\n| | hye | `$tongue$`, `$tongues$`, `$language` |\n| [1ex] | eng | `$birds$`, `$bird$` |\n| | spa | `$birds$`, `$bird$`, `$fowls$` |\n| | ell | `$birds$`, `$bird$`, `$bird` |\n| | rus | `$birds$`, `$bird$`, `$fowl`, `$bird` |\n| | tgl | `$birds$`, `$bird$`, `$fowl` |\n| | swh | `$birds$`, `$bird$`, `$fowl` |\n| | hye | `$birds$`, `$bird$`, `$flying$`, `$fowl` |\n| | | |\n| --- | --- | --- |\n| | eng | `$blood$` |\n| | deu | `$blood$` |\n| | fra | `$blood$` |\n| | jpn | `$blood$`, `$blood` |\n| | kor | `$blood$`, `$escap`, `$skin$`, `$airs$`, `$pip`, `$flut` |\n| | zho | `$blood$`, `$blood` |\n| [1ex] | eng | `$tongue$`, `$tongues$` |\n| | deu | `$tongue$`, `$tongues$` |\n| | fra | `$tongue$`, `$tongues$`, `$language` |\n| | jpn | `$tongue$`, `$tongues$` |\n| | kor | `$tongue$`, `$tongues$`, `$language` |\n| | zho | `$tongue$`, `$tongues$`, `$language` |\n| [1ex] | eng | `$mouth$`, `$mouths$` |\n| | deu | `$mouth$`, `$mouths$` |\n| | fra | `$mouth$`, `$mouths$` |\n| | jpn | `$mouth$`, `$mouths$`, `$entrance`, `$kiss`, `$whistl`, `$doorkeeper`, `$contention` |\n| | kor | `$mouth$`, `$mouths$`, `$entrance`, `$lip`, `$clothe`, `$kiss`, `$overla` |\n| | zho | `$mouth$`, `$mouths$`, `$entrance`, `alat`, `$ford` |\n| | | |\n| --- | --- | --- |\n| | spa | `$birds$`, `$bird$`, `$fowls$` |\n| | tgl | `$birds$`, `$bird$`, `$fowl` |\n| | ceb | `$birds$`, `$bird$`, `$fowl` |\n| | hil | `$birds$`, `$bird$`, `$sparrows$` |\n| [1ex] | spa | `$ear$`, `$ears$`, `$heard$` |\n| | tgl | `$ears$`, `$ear$`, `$listen$`, `$hearing$` |\n| | ceb | `$ears$`, `$ear$`, `$hear$`, `$hearing$` |\n| | hil | `$ear$`, `$ears$`, `$hear` |\n| [1ex] | spa | `$tongue$`, `$tongues$`, `$language` |\n| | tgl | `$tongue$`, `$tongues$`, `$language` |\n| | ceb | `$tongue$`, `$tongues$`, `$language` |\n| | hil | `$tongue$`, `$tongues$` |\n| | | |\n| --- | --- | --- |\n| | yor | `$tree$`, `$trees$`, `wood`, `$stake`, `$frankincense$`, `$thornbush`, `$palm-tree$` |\n| | ibo | `$tree$`, `$trees$`, `$pole`, `wood`, `$impal`, `$stake`, `$panel` |\n| | mcn | `$tree$`, `$trees$`, `wood`, `$stake`, `$impale`, `$cedar`, `$timber` |\n| | twi | `$tree$`, `$trees$`, `$wood`, `$panel$`, `$pole`, `$figs$`, `$timber` |\n| [1ex] | yor | `$hair$`, `$hairs$`, `$wool$` |\n| | ibo | `$hair$`, `$hairs$`, `$wool`, `$shear`, `$beard` |\n| | mcn | `$hair$`, `$hairs$`, `$wool$`, `$shave`, `$baldness$`, `$shear`, `goat` |\n| | twi | `$hair$`, `$hairs$`, `$beard`, `$shave`, `$head$`, `$wool` |\n| [1ex] | yor | `$mouth$`, `$mouths$`, `$entrance`, `$kiss`, `$palate$`, `$marvel`, `$suckling` |\n| | ibo | `$mouth$`, `$mouths$`, `$gate`, `$entrance`, `$lip`, `curse`, `$precious$` |\n| | mcn | `$mouth$`, `$mouths$`, `$lips$`, `fulfill`, `$denie`, `$disown`, `$entrance` |\n| | twi | `$mouth$`, `$mouths$`, `$gat`, `$collect`, `$lip`, `$entrance`, `$registered$` |\n| | | |\n| --- | --- | --- |\n| | cak | `$tree$`, `$trees$`, `wood`, `$pole`, `$cedar`, `$figs$`, `$palm-tree$` |\n| | kjb | `$tree$`, `$trees$`, `$cedar`, `$panel`, `$wood`, `$figs$`, `$pole` |\n| | tzj | `$tree$`, `$trees$` |\n| [1ex] | cak | `$seed$`, `$seeds$`, `braham`, `$vine`, `$sow`, `fruit`, `$harvest` |\n| | kjb | `$seed$`, `$seeds$`, `braham`, `$garden`, `fruits$`, `$vine`, `$harvest` |\n| | tzj | `$seed$`, `$seeds$`, `$sow`, `$harvest$`, `fruit` |\n| [1ex] | cak | `$knees$`, `$bow`, `$worship`, `$trembl`, `fell$` |\n| | kjb | `$knees$`, `$knee$`, `$obeisance$`, `$worship`, `$bow`, `$fell$` |\n| | tzj | `$knees$`, `$worship`, `$obeisance$`, `$fell$` |\nccll & & \n & jpn &\nUTF8bsmi口\n(mouth, opening, entrance) \n& kor &\nUTF8mj구 (口)\n(entrance, gate, mouth) \n& zho &\nUTF8bsmi口 (mouth, gate, entrance), 嘴 (mouth, lips)\n \n& fra & bouche (mouth) \n[1ex] & spa & lengua (tongue, language) \n& tgl & dilà (tongue, language), wikà (tongue, language) \n& ceb & dila (tongue), pinulongan (tongue, language) \n& hil & dilà (tongue), dilâ (tongue) \n& msa & lidah (tongue), oojoo leeda (tongue) \n[1ex]\n---\n1. We release our code at [https://github.com/yihongL1U/conceptualizer]( https://github.com/yihongL1U/conceptualizer)[↩](#fnref1)\n2. For some focal concepts that are less divergent, e.g.,, we obtain fewer than 99 dimensions[↩](#fnref2)\n3. https://iso639-3.sil.org/[↩](#fnref3)\n4. https://translate.panlex.org/[↩](#fnref4)\n5. https://norare.clld.org/[↩](#fnref5)\n6. https://www.python.org/[↩](#fnref6)\n7. https://numpy.org/[↩](#fnref7)\n8. https://scipy.org/[↩](#fnref8)\n9. https://scikit-learn.org/stable/[↩](#fnref9)\n10. https://networkx.org/[↩](#fnref10)\n11. https://matplotlib.org/[↩](#fnref11)\n"},"data_id":{"kind":"string","value":"arxiv_0182802"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":456,"numItemsPerPage":100,"numTotalItems":46503,"offset":45600,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjIyMTcwNywianRpIjoiZDQ2ODg5NzgtZmI5NC00NGVkLWI5YjAtMWQ4ODU1NWUzYzQ3Iiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYyMjUzMDcsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.9Ame0eOyGauFGfFlI_adGCqDtYLFrPtV61IX78qzIUfyHn3Ysre4kO-9SPwJGRa2txyM_5Fe_1_O7ooA-5HWCQ","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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, our evaporation model includes significant simplifications. In reality, the evaporation efficiency that characterizes the how much heating energy is converted to *P*d*V* work depends on the characteristics of a planet and changes with time. Moreover, the direct transition from an X-ray regime to an EUV-driven regime,... | arxiv_0182413 |
EmptyHeaded: A Relational Engine for Graph Processing
=====================================================
There are two types of high-performance graph processing engines: low- and high-level engines. Low-level engines (Galois, PowerGraph, Snap) provide optimized data structures and computation models but require use... | arxiv_0182416 |
Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives
=====================================================================================
We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fund... | arxiv_0182421 |
Anomalous dynamics in the ergodic side of the Many-Body Localization transition and the glassy phase of Directed Polymers in Random Media
=========================================================================================================================================
Using the non-interacting Anderson tight-bin... | arxiv_0182427 |
Neumann Domains on Graphs and Manifolds
=======================================
#1
#1(#1)
#1#2
The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points ... | arxiv_0182431 |
Introduction
============
A key question in high-energy physics is whether the observed quarks and leptons are elementary or composite particles. Within the last three decades little progress has been made towards answering this important question. One reason is that experimental searches for quark or lepton substructu... | arxiv_0182435 |
Solvable Lattice Hamiltonians with Fractional Hall Conductivity
===============================================================
We construct a class of lattice Hamiltonians that exhibit fractional Hall conductivity. These Hamiltonians, while not being exactly solvable, can be controllably solved in their low energy sec... | arxiv_0182437 |
Sudakov Double Logarithms Resummation in Hard Processes in Small-*x* Saturation Formalism
=========================================================================================
In this manuscript, we present a complete study of the Sudakov double logarithms resummation for various hard processes in *e**A* and *p**A*... | arxiv_0182442 |
The MASIV Survey IV: relationship between intra-day scintillation and intrinsic variability of radio AGNs
=========================================================================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
We investigate the relationship between 5GH... | arxiv_0182445 |
�*S*5 of the CARAT IDs (5, 911, 4) and (5, 946, 2)][exq5]
```
gap> G:=CaratMatGroupZClass(5,911,4);; # G=S5
gap> Rank(FlabbyResolution(G).actionF.1); # F is of rank 11
11
gap> mis:=SearchCoflabbyResolutionBase(TransposedMatrixGroup(G),3);; # Method III
gap> List(mis,Length);
[ 36, 16, 16, 16, 16, 6 ]
gap> mi:=mis[L... | arxiv_0182451 |
Optimally Controlling Nutrition and Propulsion Force in a Long Distance Running Race[5](#fn5)
=============================================================================================
Runners competing in races are looking to optimize their performance. In this paper, a runner’s performance in a race, such as a mar... | arxiv_0182452 |
[
0.3in ]
Since the objective functions of reinforcement learning problems are typically highly nonconvex, it is desirable that policy gradient, the most popular algorithm, escapes saddle points and arrives at second-order stationary points. Existing results only consider vanilla policy gradient algorithms with unbiase... | arxiv_0182453 |
Dynamical Density Functional Theory For Microswimmers
=====================================================
Dynamical density functional theory (DDFT) has been successfully derived and applied to describe on the one hand passive colloidal suspensions, including hydrodynamic interactions between individual particles. On... | arxiv_0182454 |
of FUV emission in TDFs validates the *β*-disk—indeed, detailed MHD simulations typically find that stress scales proportionally to total pressure - our models seem to demonstrate two features of late-time TDF disk hydrodynamics. First, these disks must have long initial viscous times. This is achieved naturally in t... | arxiv_0182471 |
Retrograde Polish Doughnuts around Boson Stars
==============================================
We investigate polish doughnuts with a uniform constant specific angular momentum distribution in the space-times of rotating boson stars. In such space-times thick tori can exhibit unique features not present in Kerr space-ti... | arxiv_0182475 |
,2)node[petitrouge]; (1,2)node[petitrouge]; (3,2)node[petitrouge]; (5,2)node[petitrouge]; (-4,3)node[petitrouge]; (-2,3)node[petitrouge]; (0,3)node[petitrouge]; (1.5,3)node[petitrouge]; (2.5,3)node[petitrouge]; (3.5,3)node[petitrouge]; (5.5,3)node[petitrouge]; (-5,4)node[petitrouge]; (-3.5,4)node[petitrouge]; (-2,4)nod... | arxiv_0182485 |
Applications of a finite-volume algorithm for incompressible MHD problems
=========================================================================
[firstpage]
We present the theory, algorithms and implementation of a parallel finite-volume algorithm for the solution of the incompressible magnetohydrodynamic (MHD) equa... | arxiv_0182494 |
Primitive forms and Frobenius structures on the Hurwitz spaces
==============================================================
The main goal of this paper is to introduce the notion of a primitive form for a generic family of Hurwitz covers of P1 with a fixed ramification profile over infinity. We prove that primitive f... | arxiv_0182497 |
Giant planet migration, disk evolution, and the origin of transitional disks
============================================================================
We present models of giant planet migration in evolving protoplanetary disks. Our disks evolve subject to viscous transport of angular momentum and photoevaporation, ... | arxiv_0182499 |
Optical response of (InGa)(AsSb)/GaAs quantum dots embedded in a GaP matrix
===========================================================================
The optical response of (InGa)(AsSb)/GaAs quantum dots (QDs) grown on GaP (001) substrates is studied by means of excitation and temperature-dependent photoluminescence... | arxiv_0182500 |
An Anderson-like model of the QCD chiral transition
===================================================
Introduction
============
The low end of the spectrum of the Euclidean Dirac operator plays an important role in determining the properties of hadronic matter in Quantum Chromodynamics (QCD). In particular, the spect... | arxiv_0182507 |
Optimal Rate-Matrix Pruning For Large-Scale Heterogeneous Systems
=================================================================
We present an analysis of large-scale load balancing systems, where the processing time distribution of tasks depends on both the task and server types. Our study focuses on the asymptotic... | arxiv_0182508 |
Cliques in geometric inhomogeneous random graphs
================================================
[thm]Lemma [thm]Corollary [thm]Proposition
[thm]Remark
Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number of cliques of any size *k* cont... | arxiv_0182509 |
., Klaas, U., & Lemke, D. 2001,, 372, 719
Miley, G., & De Breuck, C. 2008,, 15, 67
Mor, R., Netzer, H., & Elitzur, M. 2009,, 705, 298
Napier, P. J., Thompson, A. R., & Ekers, R. D. 1983, IEEE Proceedings, 71, 1295
Nenkova, M., Ivezić, Ž., & Elitzur, M. 2002,, 570, L9
Nesvadba, N. P. H., et al. 2009,, 395, L16
Nesvadba,... | arxiv_0182513 |
Analytic solution and stationary phase approximation for the Bayesian lasso and elastic net
===========================================================================================
The lasso and elastic net linear regression models impose a double-exponential prior distribution on the model parameters to achieve reg... | arxiv_0182514 |
Defining and estimating causal direct and indirect effects when setting the mediator to specific values is not feasible
=======================================================================================================================
Natural direct and indirect effects decompose the effect of a treatment into the... | arxiv_0182523 |
not forbidden. For the Ising regime of the XXZ model, however, a finite lower bound on the diffusion constant has been found recently, thus ruling out subdiffusive spreading. We find the correct picture may involve another peak forming on much lower frequency scales, whose properties cannot be gleaned within the simpl... | arxiv_0182525 |
Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner
==========================================================================
We apply a method inspired by Ratner’s work on quantitative mixing for the geodesic flow (Ergod. Theory Dyn. Syst., 1987) and developed by Burger (Duke Math. J., 1990) to... | arxiv_0182534 |
fide* AGN & & C & 12.51 & 42.4 & > 1.0 & 0.43 & 424 &
19 & 104.56145 & -55.93076 & 0.298 & 2 & IRAC & ALG & & & A & 3.081 & & &
20 & 104.56586 & -56.03525 & 0.299 & 3 & IRAC & ELG & & *Candidate* AGN & B & 5.85 & & & & & 7
21 & 104.60170 & -55.82924 & 0.289 & 3 & IRAC & ELG & & Galaxy & B & 7.345 & & & & &9
2... | arxiv_0182536 |
Brownian motion on stable looptrees
===================================
In this article, we introduce Brownian motion on *α*-stable looptrees using resistance techniques, where *α* ∈ (1, 2). We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise ... | arxiv_0182542 |
Energy-Efficient UAV Communication with Trajectory Optimization
===============================================================
Wireless communication with unmanned aerial vehicles (UAVs) is a promising technology for future communication systems. In this paper, we study energy-efficient UAV communication with a ground... | arxiv_0182546 |
G*), is the smallest width of a connected path-decomposition of *G*.
Let us notice that if P = ⟨*A*1, …, *A*ℓ⟩ is a path-decomposition of a graph *G*, then Pʹ = ⟨*A**p*, …, *A*1⟩ is also a path-decomposition of *G*. But the fact that P is a connected path-decomposition does not imply that Pʹ is a connected path-decompo... | arxiv_0182558 |
mount University. The work of PN has been supported by the German Research Foundation (DFG) grant NI 1282/2-1, partially by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer Exzellenz) launched by the State of Hesse and p... | arxiv_0182561 |
are the hybridized atom-light modes which have the strongest atom-light coupling and tend to lie lowest in the spectrum. Since the hybridized modes contain both atomic and light components, their occupation number is not exactly equal to the atomic fluctuation occupation number. Nonetheless, in this system the atom-li... | arxiv_0182563 |
⋅ 4*n* − 2; see for a specific formula for calculating the genus of surfaces constructed from rational billiards. This implies that the sequence {*g**n*}*n* = 0∞ diverges to + ∞. We conjecture that there exists a surface S(*K**S*) that can be appropriately viewed as a suitable limit of the flat surfaces S(*K**S**n*) ... | arxiv_0182567 |
Market Self-Learning of Signals, Impact and Optimal Trading: 0.5cm Invisible Hand Inference with Free Energy 0.7cm (or, How We Learned to Stop Worrying and Love Bounded Rationality)
1.0cm Igor Halperin and Ilya Feldshteyn
0.5cm NYU Tandon School of Engineering
0.5cm e-mail: igor.halperin@nyu.edu, if@q-rd.com 0.5c... | arxiv_0182570 |
w() & > & |w1() I1
[cond] |w() & > & |wi() Ii Ii-1 2i N-k, where *w̄*crit*i*(*τ*) is defined by ([critcurve]).
Since *w̄*(*τ*) = *w̄*base*i*(*τ*) + *o*(1), it follows that *w̄*base*i*(*τ*) also satisfies ([cond]) if *w̄*(*τ*) satisfies condition ([cond]). It can be easily verified from ([wavg]) that if *w̄... | arxiv_0182574 |
A Reallocation Algorithm for Online Split Packing of Circles
============================================================
The Split Packing algorithm is an offline algorithm that packs a set of circles into triangles and squares up to critical density. In this paper, we develop an online alternative to Split Packing t... | arxiv_0182579 |
�*r*, *i* = 22.4, 20.0, 17.4) than the typical FGKM main sequence star targeted with Kepler, the achieved precision of 7 mmags per observation compares favorably with ground-based data. We report on the Kepler photometry and supporting multi-wavelength observations in Section [da], discuss periodicity in Section [sec-p... | arxiv_0182582 |
1 of the generator L to exist. Clearly, it can be well defined only on a subspace that does not contain elements of the kernel of L. To this purpose, consider the operator $\mathcal{F}:={\rm id}-\mathcal{G}$. Notice that
L ∘ F = F ∘ L,
so that the range of F, ${\rm Ran}(\mathcal{F})$, is mapped into itself by the... | arxiv_0182587 |
uon beam line of the Fermilab Accelerator Complex . A schematic layout of the experiment is shown in Figure [Fig:SeaQuest]. A high-intensity beam of 120 GeV protons (center of mass energy $\sqrt s\simeq15$ GeV) is delivered to a thin nuclear target. The target is situated ∼ 1 m upstream of a 5 m long, closed-aperture,... | arxiv_0182598 |
Groundstatable fermionic wavefunctions and their associated many-body Hamiltonians
==================================================================================
.
In the vast majority of many-body problems, it is the kinetic energy part of the Hamiltonian that is best known microscopically, and it is the detailed ... | arxiv_0182599 |
Strong Coordination over Noisy Channels
=======================================
We study the problem of strong coordination of the actions of two nodes $\mathsf X$ and $\mathsf Y$ that communicate over a discrete memoryless channel (DMC) such that the actions follow a prescribed joint probability distribution. We propo... | arxiv_0182600 |
physics in this situation due to the presence of a flat band. Hence, we employ two methods – the variational wavefunction and the EOM to examine the physical properties of this system. These two approaches are, in fact, complementary to each other.
Actually, there are two types of three-band touching fermion systems i... | arxiv_0182602 |
Experimental Quantum Enhanced Optical Interferometry
====================================================
Optical quantum interferometry represents the oldest example of quantum metrology and it is at the source of quantum technologies. The original squeezed state scheme is now a significant element of the last version... | arxiv_0182604 |
*) ⋅ *w* such that *w* is a (depth-minimal) position of *y* in the redex. As the applied rule is not pseudo-collapsing, if (*t*, *p*(*i*))*g* ≠ (*t*, *p*(*i* + 1))*g* then we also must have (*u*, *q*(*i*))*g* ≠ (*u*, *q*(*i* + 1))*g*.
The transitive closure clearly preserves the property. Transfinite reductions do not ... | arxiv_0182610 |
Freeze-in and freeze-out of sterile neutrino dark matter
========================================================
=.08em =.05em =.03em =.65ex =.4ex ==.5em =.5em
0.7cm
A sterile neutrino with a keV-scale mass is a compelling dark matter candidate. We propose a new production mechanism involving the decay and annihilatio... | arxiv_0182615 |
.5,2)
(-2.5,-1)(-2,0) (-1,-.5)(-1.5,-1.5) (0,-1)(-.5,-2) (0,-1)(.5,-2) (1,-1)(.5,-2) (1,-1)(1.5,-2) (2,-1)(1.5,-2) (3,-1.5)(2.5,-2.5)
(4,-2)(3.5,-3) (3,-1.5)(2.5,-2.5)
(-1.5,2)(-1.5,2.5) (-.5,1.5)(-.5,2.5) (-.5,2.5)(-.25,3)
(.5,1)(.5,2) (1.5,1)(1.5,2) (2.5,1)(2.5,2) (3.5,.5)(3.5,1.5) (4.5,0)(4.5,1)
(-.5,-2)(-.5,-3) (.5... | arxiv_0182618 |
AGN feedback, quiescence and CGM metal enrichment in early-type galaxies
========================================================================
### Accepted???. Received???; in original form???
[firstpage]
We present three-dimensional hydrodynamical simulations showing the effect of kinetic and radiative AGN feedback... | arxiv_0182629 |
account for finite-nuclear-size effects , a screening approximation based on the Welton interpretation and implemented in GRASP2K by Lowe et al. , is used to evaluate the SE contribution.
* QED - M3: A model QED operator, which also includes the non-local QED part to calculate the SE corrections for many-electron ato... | arxiv_0182634 |
), *η**c*(1*S*)*K*(2*S*) and *ψ*(2*S*)*K*\*(1*S*) channels are generally well aligned along their corresponding threshold lines. In particular, with a complex angle varied from 0∘ to 6∘, these energy poles are basically moving along the theoretical cut lines, and this fact confirms their nature as scattering states. In... | arxiv_0182638 |
Gapless edge states of *B**F* field theory and translation-symmetric Z2 spin liquids
====================================================================================
We study possible gapless edge states of translation-symmetric Z2 spin liquids. The gapless edge states emerge from dangling Majorana fermions at the ... | arxiv_0182640 |
Multi-Instance Partial-Label Learning: Towards Exploiting Dual Inexact Supervision
==================================================================================
Weakly supervised machine learning algorithms are able to learn from ambiguous samples or labels, e.g., multi-instance learning or partial-label learning.... | arxiv_0182645 |
Flowing Funnels: Heat sources for field theories and the AdS3 dual of CFT2 Hawking radiation
============================================================================================
We construct the general 2+1 dimensional asymptotically AdS3 solution dual to a stationary 1+1 CFT state on a black hole background. T... | arxiv_0182648 |
Gauss images of hyperbolic cuspswith convex polyhedral boundary
===============================================================
### August 13, 2009
[section] [dfn]Theorem [dfn]Proposition [dfn]Lemma [dfn]Corollary
[dfn]Remark [dfn]Example
[1](#fn1)
[2](#fn2)
We prove that a 3–dimensional hyperbolic cusp with convex pol... | arxiv_0182649 |
Gauging the cosmic microwave background
=======================================
We provide a new derivation of the anisotropies of the cosmic microwave background (CMB), and find an exact expression that can be readily expanded perturbatively. Close attention is paid to gauge issues, with the motivation to examine the ... | arxiv_0182650 |
to *z* ≈ 3 is less eminent.
Introduction
============
The formation and evolution of galaxies rank among the big questions in astronomy and still await a complete explanation. According to current theory, the formation of dark matter haloes by gravitational instabilities is an essential first step in the formation of ... | arxiv_0182653 |
Nanometre-scale probing of spin waves using single electron spins
=================================================================
**Correlated-electron systems support a wealth of magnetic excitations, ranging from conventional spin waves to exotic fractional excitations in low-dimensional or geometrically-frustrated... | arxiv_0182657 |
Antiscreening in perturbative quantum gravity and resolving the Newtonian singularity
=====================================================================================
Preprint numbers: ITP-UU-12/32, SPIN-12/30
We calculate the quantum corrections to the Newtonian potential induced by a massless, nonminimally coupl... | arxiv_0182661 |
Acceleration of charged particles due to chaotic scattering in the combined black hole gravitational field and asymptotically uniform magnetic field
====================================================================================================================================================
To test the role of la... | arxiv_0182663 |
Photoionizaton of Pure and Doped Helium Nanodroplets
====================================================
Helium nanodroplets, commonly regarded as the “nearly ideal spectroscopic matrix”, are being actively studied for more than two decades now. While they mostly serve as cold, weakly perturbing and transparent medium... | arxiv_0182667 |
search-based techniques to reduce timeouts due to constraint complexity may be an interesting line of future work, although only adequate for testing, and not verification.
Much work has been done on the complementary goal of statically determining the accuracy of floating-point computations. Fluctuat and PRECiSA es... | arxiv_0182672 |
\propto -\cos\phi\_{\bf k}$ and $d\_y({\bf
k})\propto \sin\phi\_{\bf k}$, which is parallel to the effective magnetic field due to the SOC [Eq. ([SOC])]. Thus, the induced triplet order parameter is stable due to the SOC. Moreover, the calculated results in Fig. [figyw5] show that the induced triplet order parameter a... | arxiv_0182675 |
& 10 50 40.3 & -54 25 15.1 & 1.2 & 1.5 & 1.1 & -25.4 & 1.1 & -25.4
16d &... & 2 & 285.90 & 4.53 & 10 51 17.2 & -54 18 59.6 & 2.7 & 6.9 & 2.4 & -23.7 & 2.6 & -23.6
16e &... & 2 & 286.00 & 5.23 & 10 54 00.2 & -53 43 58.5 & 1.6 & 2.0 & 1.2 & -24.5 & 1.5 & -24.2
16f &... & 2 & 286.00 & 5.07 & 10 53 30.5 & -53 52 57.... | arxiv_0182676 |
Coarse geometry of the Cops and robber game
============================================
We introduce two variations of the cops and robber game on graphs. These games yield two invariants in Z+ ∪ {∞} for any connected graph Γ, the *weak cop number $\operatorname{\mathsf{wCop}}(\Gamma)$* and *strong cop number $\opera... | arxiv_0182679 |
Near-IR Integral Field Spectroscopy study of the Star Formation and AGN of the LIRG NGC5135
===========================================================================================
We present a study of the central $\rm 2.3\,kpc$ of NGC5135, a nearby Luminous Infrared Galaxy (LIRG) with an AGN and circumnuclear star... | arxiv_0182680 |
${\mu}^1$};
\node at (0,2) {${\mu}^{k-1}$};
\node at (0,3) {${\mu}^k$};
\node[right] at (4,3) {{\hbox to 5pt{\hss ${\mu}^{k+\ell}$\hss}}};
\draw[->,short] (0,0) -- (0,1) node [midway, left] {$w\_1$};
\draw[dotted,short] (0,1) -- (0,2) node {};
\draw[->,short] (0,2) -- (0,3) node [midway, left] {$w\_k$};
\draw[->... | arxiv_0182682 |
Two-Loop QCD Corrections to the Higgs plus three-parton amplitudes with Top Mass Correction
===========================================================================================
Introduction
============
Scattering amplitudes play an indispensable role in particle physics. They act as a bridge between theory and ... | arxiv_0182690 |
{\mu}}^{\*} V\_{j \overline{\mu}} \right)\end{aligned}$$
and
$$\begin{aligned}
\label{kappa-EFA}
{\kappa}\_{ij}^{(\mu,EFA)} &=&
\left(V^{\*}U^{T} \right)\_{ij}
+\frac{1}{2}
\left(U\_{i \mu} V\_{j \mu}^{\*} - V\_{i \mu}^{\*} U\_{j \mu} \right)
\nonumber\\
&+&
\frac{1}{2}
\left(U\_{i \overline{\mu}} V\_{j \overlin... | arxiv_0182693 |
**Notes on well-distributed minimal sub-BIBDs for *λ* = 1**
Daniele Dona[1](#fn1)
Mathematisches Institut, Georg-August-Universität Göttingen
Bunsenstraße 3-5, 37073 Göttingen, Germany
`daniele.dona@mathematik.uni-goettingen.de`
110mm **Abstract.** In these notes we investigate BIBDs with *λ* = 1 that present subdesign... | arxiv_0182695 |
cf. definition of $\mathbf F\_N$ in ). In the case *λ* = 0 this follows immediately from the second condition in in the case *λ* = *μ* = 1 this follows from the last line in. In the complementary case *λ* = 1, *μ* = 0 (which implies *ν* = 0) only *D* − 1 weights can be absorbed into the energy and one weight $\hat{\mat... | arxiv_0182697 |
The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups
================================================================================
The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks d... | arxiv_0182699 |
A trion in a magnetic field revisited
=====================================
We revisit the problem of a two dimensional trion in an external magnetic field. We demonstrate that the approximations used previously for finding the energy spectrum of this system break down in the experimentally accessible range of magnetic... | arxiv_0182701 |
around 1.4 *M*⊙ and for this reason this value is taken as the neutron star canonical mass. This also led to the idea that white dwarf collapse may be an additional mechanism for the formation of neutron stars .
The neutron star radius is not measured directly, but observational data allow for indirect inference. Obs... | arxiv_0182702 |
Minimax estimation of qubit states with Bures risk
==================================================
The central problem of quantum statistics is to devise measurement schemes for the estimation of an unknown state, given an ensemble of *n* independent identically prepared systems. For locally quadratic loss functions... | arxiv_0182705 |
Blockchain Testing: Challenges, Techniques, and Research Directions
===================================================================
Specific testing solutions targeting blockchain-based software are gaining huge attention as blockchain technologies are being increasingly incorporated into enterprise systems. As blo... | arxiv_0182712 |
Is the dark-matter halo spin a predictor of galaxy spin and size?
==================================================================
[firstpage]
The similarity between the distributions of spins for galaxies ($\lamgal$) and for dark-matter haloes ($\lamh$), indicated both by simulations and observations, is naively in... | arxiv_0182714 |
Multiple valued Jacobi fields
=============================
We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold Σ of a Riemannian manifold M. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second varia... | arxiv_0182718 |
Heteronuclear long-range Rydberg molecules
==========================================
We present the formation of homonuclear Cs2, K2, and heteronuclear CsK long-range Rydberg molecules in a dual-species magneto-optical trap for 39K and 133Cs by one-photon UV photoassociation. The different ground-state-density depende... | arxiv_0182720 |
π*, is the cut-locus of *p*+ and corresponds to the south pole.
[c].46[FMEx1-L3L4]
[c].46[FMEx1-L5L6]
Figures [FMEx1-L3L4] *resp.* [FMEx1-L5L6] illustrate Proposition [Ex1-P] in the cases ℓ = 3 ... | arxiv_0182723 |
Masses and electric charges: gauge anomalies and anomalous thresholds
=====================================================================
We work out in the forward limit and up to order *e*6 in perturbation theory the collinear divergences. In this kinematical regime we discover new collinear divergences that we arg... | arxiv_0182724 |
Verification-Aided Deep Ensemble Selection
==========================================
Deep neural networks (DNNs) have become the technology of choice for realizing a variety of complex tasks. However, as highlighted by many recent studies, even an imperceptible perturbation to a correctly classified input can lead to ... | arxiv_0182729 |
Decentralized Abstractions For Multi-Agent Systems Under Coupled Constraints
============================================================================
The goal of this report is to define abstractions for multi-agent systems with feedback interconnection in their dynamics. In the proposed decentralized framework, we... | arxiv_0182730 |
idea is to borrow information from ‘neighboring’ networks (in time or with similar traits) rather than neighboring nodes due to *m* being much larger than *n*2.
2. If *m* ≪ (*n**ρ**n*2)− 1/2, then one should estimate a graphon *f̂* for each observed network separately (using an existing approach for single networks, e... | arxiv_0182733 |
Modeling Heterogeneous Network Interference Using Poisson Point Processes
=========================================================================
Cellular systems are becoming more heterogeneous with the introduction of low power nodes including femtocells, relays, and distributed antennas. Unfortunately, the resulti... | arxiv_0182737 |
The COSMOS AGN Spectroscopic Survey I: XMM Counterparts
=======================================================
We present optical spectroscopy for an X-ray and optical flux-limited sample of 677 XMM-Newton selected targets covering the 2 deg2 COSMOS field, with a yield of 485 high-confidence redshifts. The majority of... | arxiv_0182741 |
Holographic solar systems and hydrogen atoms:non-relativistic physics in AdS and its CFT dual
=============================================================================================
We study a non-relativistic limit of physics in AdS which retains the curvature through a harmonic Newtonian potential. This limit a... | arxiv_0182747 |
THE VELOCITY DISPERSION FUNCTION OF VERY MASSIVE GALAXY CLUSTERS: ABELL 2029 AND COMA
=====================================================================================
Based on an extensive redshift survey for galaxy cluster Abell 2029 and Coma, we measure the luminosity functions (LFs), stellar mass functions (SMF... | arxiv_0182748 |
continuous and always has a singular component.
Sinai [personal communication] raised the question: to find the local asymptotics for the spectral measures of translation flows. In we developed an approach to this problem and succeeded in obtaining Hölder estimates for spectral measures in the case of surfaces of gen... | arxiv_0182762 |
**Periodic solutions for a class of nonlinear partial differential equations in higher dimension**
==================================================================================================
We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonline... | arxiv_0182763 |
**A Cosmological Higgs Collider**
==================================
=1
The quantum fluctuations of the Higgs field during inflation could be a source of primordial density perturbations through Higgs-dependent inflaton decay. By measuring primordial non-Gaussianities, this so-called Higgs-modulated reheating scenario... | arxiv_0182765 |
violins. The left plot shows the addition of CURN to the M1 and M2 noise models. We notice the slight drop in the amplitude at the lowest frequency and slightly better constraint of amplitude at the second frequency bin. The free spectrum for GWB (right plot) shows a slight drop in power at the lowest frequency. These... | arxiv_0182766 |
Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loops
==================================================================================================
Bott–Cattaneo’s theory defines the integral invariants of framed rational homology 3-spheres with acyclic orthogonal local... | arxiv_0182771 |
On the analytic systole of Riemannian surfaces of finite type
=============================================================
[1](#fn1)
In we introduced, for a Riemannian surface *S*, the quantity Λ(*S*) : = inf*F**λ*0(*F*), where *λ*0(*F*) denotes the first Dirichlet eigenvalue of *F* and the infimum is taken over all... | arxiv_0182772 |
Fast and Compact Self-Stabilizing Verification, Computation,and Fault Detection of an MST
=========================================================================================
This paper demonstrates the usefulness of distributed local verification of proofs, as a tool for the design of self-stabilizing algorithms.... | arxiv_0182774 |
Reinforcement Learning and Deep Stochastic Optimal Control for Final Quadratic Hedging
======================================================================================
=1
We consider two data driven approaches, Reinforcement Learning (RL) and Deep Trajectory-based Stochastic Optimal Control (DTSOC) for hedging a ... | arxiv_0182775 |
the presence of the twist thus has an additional factor exp(*i**φ*/*N**s* ⋅ ∑*x**m**x*, 1) = exp(*i**φ* *w**s*[*m*]), where *w**s*[*m*] is the total spatial winding number of *m* in each configuration. In the definition of the spin stiffness *σ*, Eq. , the second derivative with respect to *φ* brings down − *w**s*[*m... | arxiv_0182785 |
by at least two orders of magnitude. We are planning a systematic search for other systems presenting these particular features.
Episode 2 is identified as a canonical long GRB which originates from the black hole formation process and lasts in arrival time from 50 s to 151 s after the trigger time. The good quality o... | arxiv_0182788 |
SN 2009E: a faint clone of SN 1987A
===================================
### Received XXXXX ; accepted XXXXX
In this paper we investigate the properties of SN 2009E, which exploded in a relatively nearby spiral galaxy (NGC 4141) and that is probably the faintest 1987A-like supernova discovered so far. We also attempt to... | arxiv_0182794 |
ability_concreteness_flip.png)*stability* measure vs. *concreteness* scores.
Analysis
========
Concept stability
-----------------
Figure [fig:stabilityconcreteness] and Table [tab:concretenessstabilitybinary] support our hypothesis that concreteness contributes to stability, as most concrete concepts tend to be stable... | arxiv_0182802 |
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