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@@ -0,0 +1,551 @@
+arXiv:2301.13417v1  [math.AG]  31 Jan 2023
+TEN COMPATIBLE POISSON BRACKETS ON P5
+VILLE NORDSTROM AND ALEXANDER POLISHCHUK
+Abstract. We give explicit formulas for ten compatible Poisson brackets on P5 found
+in [3].
+1. Introduction
+The goal of this paper is to present explicit formulas for certain algebraic Poisson brackets
+on P5.
+Recall that two Poisson brackets {⋅,⋅}1, {⋅,⋅}2 are called compatible if any linear combi-
+nation {⋅,⋅}1 +λ ⋅ {⋅,⋅}2 is still a Poisson bracket (i.e., satisfies the Jacobi identity). Pairs of
+compatible Poisson bracket play an important role in the theory of integrable systems.
+With every normal elliptic curve C in Pn one can associate naturally a Poisson bracket
+on Pn, called Feigin-Odesskii bracket of type qn+1,1. The corresponding quadratic Poisson
+brackets on An+1 arise as quasi-classical limit of Feigin-Odesskii elliptic algebras. On the
+other hand, they can be constructed using the geometry of vector bundles on C (see [2], [7]).
+It was discovered by Odesskii-Wolf [5] that for every n there exists a family of 9 linearly
+independent mutually compatible Poisson brackets on Pn, such that their generic linear
+combinations are Feigin-Odesskii brackets of type qn+1,1. In [3] this construction was ex-
+plained and extended in terms of anticanonical line bundles on del Pezzo surfaces. It was
+observed in [3, Ex. 4.6] that in this framework one also obtains 10 linearly independent mu-
+tually compatible Poisson brackets on P5. In this paper we will produce explicit formulas
+for these 10 brackets (see Theorem 3.2).
+2. Homological perturbation for Pn
+2.1. Formula for the homotopy. Let
+H =
+⊕
+p≥0,q∈Z
+Hp(Pn,O(q))
+be the cohomology algebra of line bundles on Pn, and
+A = ( ⊕
+p≥0,q∈Z
+Cp(Pn,O(q)),d)
+the Cech complex with respect to the standard open covering Ui = (xi ≠ 0) of Pn. There is
+a natural dg-algebra structure on A, such that the corresponding cohomology algebra is H.
+A.P. is partially supported by the NSF grant DMS-2001224, and within the framework of the HSE
+University Basic Research Program and by the Russian Academic Excellence Project ‘5-100’.
+1
+
+2
+VILLE NORDSTROM AND ALEXANDER POLISHCHUK
+The multiplication on A is defined as follows. For α ∈ Cp(Pn,O(q)) and β ∈ Cp′(Pn,O(q′))
+we define αβ ∈ Cp+p′(Pn,O(q + q′)) by
+(αβ)i0i1...ip+p′ ∶= αi0...ip∣Ui0...ip+p′ ⋅ βip...ip+p′∣Ui0...ip+p′
+where on the right hand side we use the multiplication map O(q) ⊗ O(q′) → O(q + q′).
+The homological perturbation lemma equips H with a minimal A∞-structure (mn),
+where m2 is the usual product on H. We will use the form of this lemma due to Kontsevich-
+Soibelman [4], which gives formulas for mn as sums over trees.
+To apply homological
+perturbation we need the following data:
+● a projection π ∶ A → H,
+● an inclusion ι ∶ H → A, and
+● a homotopy Q such that πι = idH and idA −ιπ = dQ + Qd.
+Recall that H0 = C[x0,...,xn],
+Hn ≃
+⊕
+e0,...,en<0
+k ⋅ xe0
+0 xe1
+1 ⋯xen
+n ⊂ An,
+and Hi = 0 for i ≠ 0,n. We define ι in degree zero by ι(f)k = f for k = 0,1,...,n. We define
+ι in degree n by ι(g)0...n = g. We define the projection in degree zero to be
+π(γ) = {γn if γn ∈ C[x0,...,xn]
+0 else.
+To define π in degree n we observe that
+An =
+⊕
+e0,...,en∈Z
+k ⋅ xe0
+0 xe1
+1 ⋯xen
+n ,
+and we let π be the natural projection to Hn.
+To define the homotopy we use that A decomposes as a direct sum of chain complexes
+A = ⊕⃗e∈Zn+1A(⃗e),
+where A(⃗e) consists of all elements in A whose components are scalar multiples of x⃗e ∶=
+xe0
+0 xe1
+1 ⋯xen
+n . In other words, A(⃗e) is the ⃗e-isotypical summand with respect to the action
+of the Gn+1
+m .
+Let us set for ⃗e ∈ Zn+1,
+k(⃗e) ∶= max{i ∣ ei ≥ 0}
+(which is equal to −∞ if all ei are negative). There is then a standard homotopy Q defined
+on an element γ ∈ A(⃗e)p by Q(γ)i0i1...ip−1 = γk(⃗e)i0...ip−1 if k(⃗e) > −∞ and Q(γ)i0i1...ip−1 = 0
+otherwise (i.e., if all ei are negative).
+For a Laurent monomial x⃗e and a subset I = {i0,...,ip} ⊂ {0,1,...,n} such that I ⊃ {0 ≤
+i ≤ n∣ei < 0}, let us denote by x⃗e
+I the element of Ap given by
+(x⃗e
+I)j0...jp = {x⃗e
+if {j0,...,jp} = I,
+0
+otherwise
+.
+
+TEN COMPATIBLE POISSON BRACKETS ON P5
+3
+Note that the condition I ⊃ {0 ≤ i ≤ n∣ei < 0} guarantees that x⃗e is a regular section of the
+appropriate line bundle over Ui0,...,ip. Clearly, these elements form a basis for A and our
+homotopy operator Q is given by
+Q(x⃗e
+I) =
+⎧⎪⎪⎨⎪⎪⎩
+(−1)jx⃗e
+I∖k(⃗e) if k(⃗e) = ij ∈ I
+0 otherwise.
+With these data one can in principle calculate all the higher products on the cohomology
+algebra H. Below we will get explicit formulas in the case we need.
+2.2. Calculation of m4 for P2. We now specialize to the case of the projective plane
+P2. We will have no higher products of odd degree because H and H⊗n only live in even
+degrees. Below we will explicitly compute the product m4. For degree reasons it will only
+be non-zero on elements e ⊗ f ⊗ g ⊗ h ∈ H⊗4 where one or two of the arguments lie in H2
+and the rest in H0. Hence, the following special case of the multiplication in A will be
+relevant: for a monomial x⃗e and a Laurent monomial x⃗e′ we have
+ι0(x⃗e) ⋅ x
+⃗e′
+I = x
+⃗e′
+I ⋅ ι0(x⃗e) = x⃗e+⃗e′
+I
+.
+We use the formula
+m4(e,f,g,h) = −∑
+T
+ǫ(T)mT (e,f,g,h)
+where the sum runs over all rooted binary trees with 4 leaves labeled e,f,g and h (from left
+to right). For each such tree T the expressions mT(e,f,g,h) is computed by moving the
+inputs through that tree, applying ι at the leaves, applying the homotopy Q on each interior
+edge, multiplying elements of A at each inner vertex and finally applying the projection π
+at the bottom.
+We have to sum over the following five trees, which we denote T1,...,T5 respectively:
+Let us first consider the case e ∈ H2 and f,g,h ∈ H0 and let’s take them all to be basis
+elements of H2 and H0:
+e = (xα0
+0 xα1
+1 xα2
+2 ){0,1,2}, f = xa0
+0 xa1
+1 xa2
+2 , g = xb0
+0 xb1
+1 xb2
+2 , h = xc0
+0 xc1
+1 xc2
+2
+where α0,α1,α2 < 0 and ai,bi,ci ≥ 0 for i = 0,1,2. In this case only one of the trees above
+can be non-zero in the expression for m4(e,f,g,h), namely T5, because in all other trees
+at some point the homotopy Q will be applied to an element of A0. Below is a picture of
+the different summands in A● and the possible ways the homotopy Q can map a monomial
+
+4
+VILLE NORDSTROM AND ALEXANDER POLISHCHUK
+element in each summand
+H2
+A2 ∶
+●0,1,2
+A1 ∶
+●0,1
+●0,2
+●1,2
+A0 ∶
+●0
+●1
+●2
+H0
+ι2
+(1)
+(3)
+(2)
+(4)
+π0
+When computing mT5(e,f,g,h) we think of it as e moving through this diagram; at every
+node it gets multiplied by one of the other arguments and then it moves downwards along
+one of the arrows. We see that to be non-zero we have to go either (1) followed by (2) or (3)
+followed by (4) (so that we land in ●2). We claim that only the second route is possible.
+The reason is because at each node we multiply e by a monomial so the exponents of
+x0,x1,x2 will not decrease at any time. By the definition of Q, if we go along (1) we must
+have that the exponent of x1 was non-negative and the exponent of x2 was negative after
+the multiplication at ●0,1,2. After performing the multiplication at ●0,2 the exponent of
+x1 will still be non-negative and it follows then from the definition of Q that (2) is not
+possible after (1).
+Now comes the computation of mT5(e,f,g,h). Below, we denote by µ the multiplication
+in A.
+mT5(e,f,g,h) =
+πµ(Qµ(Qµ(e,f),g),h) =
+πµ(Qµ(Q(xα0+a0
+0
+xα1+a1
+1
+xα2+a2
+2
+){0,1,2},g),h)
+(∗)=
+πµ(Qµ((xα0+a0
+0
+xα1+a1
+1
+xα2+a2
+2
+){1,2},g),h) =
+πµ(Q(xα0+a0+b0
+0
+xα1+a1+b1
+1
+xα2+a2+b2
+2
+){1,2},h)
+(∗∗)
+=
+π(µ((xα0+a0+b0
+0
+xα1+a1+b1
+1
+xα2+a2+b2
+2
+){2},h)) =
+π((xα0+a0+b0+c0
+0
+xα1+a1+b1+c1
+1
+xα2+a2+b2+c2
+2
+){2})
+(∗∗∗)
+=
+xα0+a0+b0+c0
+0
+xα1+a1+b1+c1
+1
+xα2+a2+b2+c2
+2
+
+TEN COMPATIBLE POISSON BRACKETS ON P5
+5
+where (∗), (∗∗) are (∗ ∗ ∗) means we get zero unless the following conditions hold
+(∗)
+⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
+α0 + a0 ≥ 0
+α1 + a1 < 0
+α2 + a2 < 0
+(∗∗) {α1 + a1 + b1 ≥ 0
+α2 + a2 + b2 < 0
+(∗ ∗ ∗)
+⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
+α0 + a0 + b0 + c0 ≥ 0
+α1 + a1 + b1 + c1 ≥ 0
+α2 + a2 + b2 + c2 ≥ 0
+.
+In the end we have
+m4(e,f,g,h) = −mT5(e,f,g,h) = −ρ(⃗α; ⃗a,⃗b, ⃗c) ⋅ x⃗α+⃗a+⃗b+⃗c,
+where
+ρ(⃗α; ⃗a,⃗b, ⃗c) ∶=
+⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
+1 if
+α0 + a0 ≥ 0
+α1 + a1 < 0
+α1 + a1 + b1 ≥ 0
+α2 + a2 + b2 < 0
+α2 + a2 + b2 + c2 ≥ 0
+0 else.
+Similarly we compute m4 applied to e,f,g,h in any given order. We have
+m4(e,f,g,h) = − ρ(⃗α; ⃗a,⃗b, ⃗c) ⋅ x⃗α+⃗a+⃗b+⃗c,
+m4(f,e,g,h) =[−ρ(⃗α; ⃗a,⃗b, ⃗c) + ρ(⃗α;⃗b, ⃗a, ⃗c) − ρ(⃗α;⃗b, ⃗c, ⃗a)] ⋅ x⃗α+⃗a+⃗b+⃗c,
+m4(f,g,e,h) =[ρ(⃗α;⃗b, ⃗a, ⃗c) − ρ(⃗α;⃗b, ⃗c, ⃗a) + ρ(⃗α; ⃗c,⃗b, ⃗a)] ⋅ x⃗α+⃗a+⃗b+⃗c,
+m4(f,g,h,e) =ρ(⃗α; ⃗c,⃗b, ⃗a) ⋅ x⃗α+⃗a+⃗b+⃗c.
+3. Feigin-Odesskii brackets
+3.1. Bivectors on projective spaces. It is well known that every Gm-invariant bivector
+on a vector space V leads to a bivector on the projective space PV . A bivector on V can be
+thought of as a skew-symmetric bracket {⋅,⋅} on the polynomial algebra S(V ∗), which is a
+biderivation. Such a bracket is Gm-invariant if and only if the bracket of two linear forms
+is a quadratic form. In other words, such a bracket can be viewed as a skew-symmetric
+pairing
+b ∶ V ∗ × V ∗ → S2(V ∗).
+
+6
+VILLE NORDSTROM AND ALEXANDER POLISHCHUK
+The corresponding bivector Π on the projective space PV is determined by the skew-
+symmetric forms Πv on T ∗
+v PV for each point ⟨v⟩ ∈ PV . We have an identification
+T ∗
+v PV = ⟨v⟩∨ ⊂ V ∗.
+It is easy to see that under this identification we have
+Πv(s1 ∧ s2) = b(s1,s2)(v),
+(3.1)
+where s1,s2 ∈ ⟨v⟩∨. Here we take the value of the quadratic form b(s1 ∧ s2) at v.
+We can use the above formula in reverse. Namely, suppose for some bivector Π on PV
+we found a skew-symmetric pairing b such that (3.1) holds. Then the Gm-invariant bracket
+{⋅,⋅} on S(V ) given by b induces the bivector Π on PV . Note that if Π is a Poisson bivector
+on PV , it is not guaranteed that the Gm-invariant bracket {⋅,⋅} on S(V ) is also Poisson,
+i.e., satisfies the Jacobi identity (but it is known that {⋅,⋅} can be chosen to be Poisson,
+see [1], [6]).
+3.2. Recollections from [3]. Let ξ be a line bundle of degree n on an elliptic curve C.
+We fix a trivialization ωC ≃ OC. Then the associated Feigin-Odesskii Poisson structure
+Π (to which we will refer as FO bracket) on PH1(ξ−1) ≃ PH0(ξ)∗ is given by the formula
+(see [3, Lem. 2.1])
+Πφ(s1 ∧ s2) = ⟨φ,MP(s1,φ,s2)⟩,
+(3.2)
+where ⟨φ⟩ ∈ PExt1(ξ,O), and s1,s2 ∈ ⟨φ⟩⊥. Here we use the Serre duality pairing ⟨⋅,⋅⟩
+between H0(ξ) and H1(ξ−1) and the triple Massey product
+MP ∶ H0(ξ) ⊗ H1(ξ−1) ⊗ H0(ξ) → H0(ξ)
+that also agrees with the triple product m3 obtained by homological perturbation from the
+natural dg enhancement of the derived category of coherent sheaves on C. There is some
+ambiguity in a choice of m3 but for s1,s2 ∈ ⟨φ⟩⊥, the expression in the right-hand side of
+(3.2) is well defined.
+Next, assume that S is a smooth projective surface, L is a line bundle on S such that
+H∗(L ⊗ KS) = 0. Then for each smooth (connected) anticanonical divisor C ⊂ S (which is
+an elliptic curve), we have a natural restriction map
+H0(S,L) → H0(C,L∣C).
+The exact sequence
+0 → LKS
+F✲ L → LC → 0
+shows that under our assumptions this restriction map is an isomorphism.
+Thus, the FO bracket on PH0(L∣C)∗ associated with (C,L∣C) (defined up to rescaling)
+can be viewed as a Poisson structure on a fixed projective space PV ∗, where
+V ∶= H0(S,L).
+By [3, Thm. 4.4], the Poisson brackets on PV ∗ associated with different anticanonical
+divisors are compatible. More precisely, we get a linear map from H0(S,K−1
+S ) to the space
+of bivectors on PV ∗, whose image lies in the space of Poisson brackets.
+
+TEN COMPATIBLE POISSON BRACKETS ON P5
+7
+3.3. Feigin-Odesskii bracket for an anticanonical divisor. We keep the data (S,L)
+of the previous subsection. Let i ∶ C ↪ S be an anticanonical divisor in S, with the equation
+F ∈ H0(S,K−1
+S ). We want to write a formula for the FO bracket Π = ΠF on PV ∗ in terms
+of higher products on the surface S and the equation F. For this we rewrite the right-hand
+side of formula (3.2). Let us write the triple product in this formula as MP C to remember
+that it is defined for the derived category of C.
+Proposition 3.1. (i) In the above situation, given e ∈ V ∗ and s1,s2 ∈ ⟨e⟩⊥, one has
+⟨e,MP C(s1∣C,e,s2∣C)⟩ = ⟨m4(F,s1,e,s2) − m4(s1,F,e,s2),e⟩,
+where we use the identification V ∗ ≃ H2(S,L−1KS) given by Serre duality and consider the
+A∞-products on S,
+m4 ∶ H0(K−1
+S )H0(L)H2(L−1KS)H0(L) → H0(L), H0(L)H0(K−1
+S )H2(L−1)H0(L) → H0(L),
+obtained by the homological perturbation.
+(ii) Assume that a generic anticanonical divisor is smooth (and connected). Then
+ΠF∣e(s1 ∧ s2) ∶= ⟨m4(F,s1,e,s2) − m4(s1,F,e,s2),e⟩,
+gives a collection of compatible Poisson brackets on PV depending linearly on F.
+Proof. (i) By Serre duality, H∗(S,L−1) = 0, so the map
+H1(C,L−1∣C) → H2(S,L−1KS),
+induced by the exact sequence
+0 → L−1KS → L−1 → L−1∣C → 0,
+is an isomorphism. It is a standard fact that this isomorphism is the dual to the isomor-
+phism H0(S,L) → H0(C,L∣C) given by the restriction, via Serre dualities on S and C. Let
+us denote by eC ∈ H1(C,L−1∣C) the element corresponding to e ∈ H2(S,L−1KS) under the
+above isomorphism.
+We claim that the triple Massey product MP C(s1∣C,eC,s2∣C) = m3(s1∣C,eC,s2∣C) corre-
+sponding to the arrows
+OC
+s2∣C✲ L∣C
+[1]✲ OC
+s1∣C✲ L∣C
+agrees with the corresponding triple Massey product on S,
+OS → L
+[1]✲ OC → L∣C.
+Indeed, the relevant spaces are identified via the restriction maps.
+Let r ∶ OS → OC,
+rL ∶ L → L∣C be the natural maps. Then we have to check that for s1,s2 ∈ ⟨e⟩⊥ ⊂ H0(S,L),
+one has
+m3(s1∣C,eC,s2∣C)r ≡ m3(s1∣C,eCrL,s2)
+mod ⟨s1∣Cr,s2∣Cr⟩,
+where we view this as equality of cosets in Hom(OS,L∣C). The A∞-identities imply that
+m3(s1∣C,eC,s2∣C)r ≡ m3(s1∣C,eC,s2∣Cr) ± s1∣Cm3(eC,s2∣C,r),
+where s2∣Cr = rLs2, and
+m3(s1∣C,eC,rLs2) = m3(s1∣C,eCrL,s2) ± s1∣Cm3(eC,rL,s2) ± m2(s1∣C,eC,rL)s2.
+
+8
+VILLE NORDSTROM AND ALEXANDER POLISHCHUK
+Combining these two identities we deduce our claim.
+Thus, it is enough to calculate the Massey product MP(s1∣C,eCrL,s2). Using the exact
+sequences above we can represent OC (resp., LC) by the twisted complex [KS[1] → OS]
+(resp., [LKS[1] → L]).
+In terms of these resolutions the elements of Ext1(L,OC) get represented by Ext2(L,KS) ⊂
+hom●(L,[KS[1] → OS]), while the element of Hom(OC,L∣C) corresponding to s ∈ H0(S,L) ≃
+H0(C,L∣C) is given by the natural map of twisted complexes induced by the multiplication
+by s. The elements of Hom(OS,L∣C) are identified with Hom(OS,L) ≃ hom0(OS,[LKS[1] →
+L]). Thus, the m3 product we are interested is given by the following triple product in the
+category of twisted complexes over S:
+OS
+L
+s2
+❄
+KS[1]
+e
+❄
+F
+✲ OS
+LKS[1]
+s1
+❄
+F
+✲ L
+s1
+❄
+where we view e as a morphism of degree 1 from L to KS[1]. Now the formula for m3 on
+twisted complexes gives
+m4(F,s1,e,s2) − m4(s1,F,e,s2).
+(ii) It is clear that ΠF gives a linear map from H0(S,ω−1
+S ) to the space of bivectors on PV .
+By (i), for generic F we get a Poisson bracket. Hence, this is true for all F.
+□
+3.4. The case leading to 10 compatible brackets on P5. We can apply Proposition
+3.1 to the case S = P2 and L = O(2). Note that the assumptions are satisfied in this case
+since LKS = O(−1) has vanishing cohomology. Thus, for each F ∈ H0(P2,O(3)) giving a
+smooth cubic, we get a formula for the FO-bracket ΠF on PH0(P2,O(2))∗ = P5. Hence,
+we get a family of 10 (the dimension of H0(P2,O(3)) compatible brackets on P5 (we also
+know this from [3, Prop. 4.7]). The fact that these 10 brackets are linearly independent
+follows from the compatibility of this construction with the GL3-action and is explained
+in [3, Prop. 4.7].
+
+TEN COMPATIBLE POISSON BRACKETS ON P5
+9
+Now we will derive formulas for the the brackets {,}F on the algebra of polynomials in
+6 variables which induce the above Poisson brackets on PV ≃ P5, where
+V = H0(P2,O(2))∗.
+They depend linearly on F, so we will just give formulas for {,}x⃗c, where x⃗c runs through
+all 10 monomials of degree 3 in (x0,x1,x2).
+Let us set
+∆(n) ∶= {{(a0,a1,a2) ∈ Z3 ∣a0 + a1 + a2 = n,ai ≥ 0 for i = 0,1,2} if n ≥ 0
+{(α0,α1,α2) ∈ Z3 ∣α0 + α1 + α2 = n,αi < 0 for i = 0,1,2} if n < 0.
+Note that {x⃗e ∣ ⃗e ∈ ∆(n)} forms a basis for H0(P2,O(n)) when n ≥ 0, while {x⃗e
+{0,1,2} ∣ ⃗e ∈
+∆(n)} is a basis for H2(P2,O(n)) when n < 0. In particular, we use {x⃗a ∣ ⃗a ∈ ∆(2)} as
+a basis in V ∗ = H0(P2,O(2)). Our brackets should associate to a pair of elements of this
+basis a quadratic form in the same variables.
+Theorem 3.2. One has for ⃗a,⃗b ∈ ∆(2), ⃗c ∈ ∆(3),
+{x⃗a,x
+⃗b}x⃗c ∶=
+∑
+⃗a′,⃗b′∈∆(2)
+[∑
+σ
+−sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗a′, ⃗b′)]x
+⃗a′x
+⃗b′
+(3.3)
+where the second sum is over the symmetric group on the letters {a,b,c} and
+˜ρ(⃗a,⃗b, ⃗c, ⃗a′, ⃗b′) ∶=
+⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
+1 if
+a′
+0 ≤ a0 − 1
+a′
+1 > a1 − 1
+a′
+1 ≤ a1 + b1 − 1
+a2 + b2 < a′
+2 + 1
+c2 + a2 + b2 ≥ a′
+2 + 1
+a′
+0 + b′
+0 = a0 + b0 + c0 − 1
+a′
+1 + b′
+1 = a1 + b1 + c1 − 1
+0 else.
+Proof. By Serre duality, we can identify V = H0(P2,O(2))∗ with H2(P2,O(−5)).
+By
+Proposition 3.1, the bracket {x⃗a,x⃗b}x⃗c is the quadratic form on V ≃ H2(P2,O(−5)) given
+by
+Q(e) ∶= ⟨e,m4(x⃗c,x⃗a,e,x
+⃗b) − m4(x⃗a,x⃗c,e,x
+⃗b)⟩.
+We can write
+e =
+∑
+⃗α∈∆(−5)
+c⃗αx⃗α
+{0,1,2} ∈ H2(P2,O(−5)).
+Using the formulas for m4 from the end of section 2.2 we get
+Q(e) =
+∑
+⃗α, ⃗β∈∆(−5)
+[∑
+σ
+−sgn(σ)ρ(⃗α;σ⃗a,σ⃗b,σ⃗c)]δ(⃗α, ⃗β, ⃗a,⃗b, ⃗c)c⃗αc ⃗β,
+
+10
+VILLE NORDSTROM AND ALEXANDER POLISHCHUK
+where the second sum runs over the symmetric group on the letters {a,b,c} and
+δ(⃗α, ⃗β, ⃗a,⃗b, ⃗c) = {1 if ⃗α + ⃗β + ⃗a + ⃗b + ⃗c = (−1,−1,−1)
+0 else.
+We have to show that the element in S2(H0(P2,O(2))) given by the right-hand side of
+(3.3) defines the same quadratic form Q on H2(P2,O(−5)). To see this we apply it to the
+element e = ∑⃗α∈∆(−5) c⃗αx⃗α
+{0,1,2} ∈ H2(P2,O(−5)). For ⃗α ∈ O(−5) we set ⃗α∗ ∶= (−1,−1,−1)− ⃗α
+and then we compute
+(
+∑
+⃗a′,⃗b′∈∆(2)
+[∑
+σ
+−sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗a′, ⃗b′)]x
+⃗a′x
+⃗b′)(e) =
+∑
+⃗α, ⃗β∈∆(−5)
+∑
+⃗a′,⃗b′∈∆(2)
+[∑
+σ
+−sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗a′, ⃗b′)]⟨x
+⃗a′,x⃗α
+{0,1,2}⟩⟨x
+⃗b′,x
+⃗β
+{0,1,2}⟩c⃗αc ⃗β =
+∑
+⃗α, ⃗β∈∆(−5)
+[∑
+σ
+−sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗α∗, ⃗β∗)]c⃗αc ⃗β.
+Now it only remains to note that ˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗α∗, ⃗β∗) = ρ(⃗α;σ⃗a,σ⃗b,σ⃗c)δ(⃗α, ⃗β, ⃗a,⃗b, ⃗c) for
+any permutation σ.
+□
+Remarks 3.3. 1. Note that when we take ⃗c = (0,0,3) only two permutations σ, namely,
+σ = 1 and σ = (a b), can give non-zero terms in the formula of Theorem 3.2.
+When
+⃗c = (1,2,0) all permutations except σ = 1 and σ = (a b) may give non-zero terms. When
+⃗c = (1,1,1) all permutations can give non-zero terms.
+2. We do not claim that formulas (3.3) define Poisson brackets and are compatible on the
+algebra of polynomials in 6 variables, only that this holds for the induced brackets on P5.
+References
+[1] A. Bondal, Non-commutative deformations and Poisson brackets on projective spaces, preprint MPI
+93-67
+[2] B. L. Feigin, A. V. Odesskii, Vector bundles on an elliptic curve and Sklyanin algebras, in Topics in
+quantum groups and finite-type invariants, 65–84, Amer. Math. Soc., Providence, RI, 1998.
+[3] Z. Hua, A. Polishchuk, Elliptic bihamiltonian structures from relative shifted Poisson structures,
+arXiv:2007.12351.
+[4] M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibrations, in Symplectic geom-
+etry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publishing, River Edge, NJ, 2001.
+[5] A. Odesskii, T. Wolf, Compatible quadratic Poisson brackets related to a family of elliptic curves,
+arXiv:1204.1299
+[6] A. Polishchuk, Algebraic geometry of Poisson brackets, Journal of Math. Sciences 84 (1997) 1413–1445.
+[7] A. Polishchuk, Poisson structures and birational morphisms associated with bundles on elliptic curves,
+IMRN 13 (1998), 683–703.
+Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
+Email address: [email protected]
+
+TEN COMPATIBLE POISSON BRACKETS ON P5
+11
+Department of Mathematics, University of Oregon, Eugene, OR 97403, USA; National
+Research University Higher School of Economics
+Email address: [email protected]
+

09FQT4oBgHgl3EQf0jbn/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf,len=293
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='13417v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='AG]  31 Jan 2023  TEN COMPATIBLE POISSON BRACKETS ON P5  VILLE NORDSTROM AND ALEXANDER POLISHCHUK  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We give explicit formulas for ten compatible Poisson brackets on P5 found  in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Introduction  The goal of this paper is to present explicit formulas for certain algebraic Poisson brackets  on P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Recall that two Poisson brackets {⋅,⋅}1, {⋅,⋅}2 are called compatible if any linear combi-  nation {⋅,⋅}1 +λ ⋅ {⋅,⋅}2 is still a Poisson bracket (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=', satisfies the Jacobi identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Pairs of  compatible Poisson bracket play an important role in the theory of integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  With every normal elliptic curve C in Pn one can associate naturally a Poisson bracket  on Pn, called Feigin-Odesskii bracket of type qn+1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' The corresponding quadratic Poisson  brackets on An+1 arise as quasi-classical limit of Feigin-Odesskii elliptic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' On the  other hand, they can be constructed using the geometry of vector bundles on C (see [2], [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  It was discovered by Odesskii-Wolf [5] that for every n there exists a family of 9 linearly  independent mutually compatible Poisson brackets on Pn, such that their generic linear  combinations are Feigin-Odesskii brackets of type qn+1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' In [3] this construction was ex-  plained and extended in terms of anticanonical line bundles on del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' It was  observed in [3, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='6] that in this framework one also obtains 10 linearly independent mu-  tually compatible Poisson brackets on P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' In this paper we will produce explicit formulas  for these 10 brackets (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Homological perturbation for Pn  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Formula for the homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Let  H =  ⊕  p≥0,q∈Z  Hp(Pn,O(q))  be the cohomology algebra of line bundles on Pn, and  A = ( ⊕  p≥0,q∈Z  Cp(Pn,O(q)),d)  the Cech complex with respect to the standard open covering Ui = (xi ≠ 0) of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' There is  a natural dg-algebra structure on A, such that the corresponding cohomology algebra is H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' is partially supported by the NSF grant DMS-2001224, and within the framework of the HSE  University Basic Research Program and by the Russian Academic Excellence Project ‘5-100’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  1  2  VILLE NORDSTROM AND ALEXANDER POLISHCHUK  The multiplication on A is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' For α ∈ Cp(Pn,O(q)) and β ∈ Cp′(Pn,O(q′))  we define αβ ∈ Cp+p′(Pn,O(q + q′)) by  (αβ)i0i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip+p′ ∶= αi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip∣Ui0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip+p′ ⋅ βip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip+p′∣Ui0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip+p′  where on the right hand side we use the multiplication map O(q) ⊗ O(q′) → O(q + q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  The homological perturbation lemma equips H with a minimal A∞-structure (mn),  where m2 is the usual product on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We will use the form of this lemma due to Kontsevich-  Soibelman [4], which gives formulas for mn as sums over trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  To apply homological  perturbation we need the following data:  a projection π ∶ A → H,  an inclusion ι ∶ H → A, and  a homotopy Q such that πι = idH and idA −ιπ = dQ + Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Recall that H0 = C[x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',xn],  Hn ≃  ⊕  e0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',en<0  k ⋅ xe0  0 xe1  1 ⋯xen  n ⊂ An,  and Hi = 0 for i ≠ 0,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We define ι in degree zero by ι(f)k = f for k = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We define  ι in degree n by ι(g)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='n = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We define the projection in degree zero to be  π(γ) = {γn if γn ∈ C[x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',xn]  0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  To define π in degree n we observe that  An =  ⊕  e0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',en∈Z  k ⋅ xe0  0 xe1  1 ⋯xen  n ,  and we let π be the natural projection to Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  To define the homotopy we use that A decomposes as a direct sum of chain complexes  A = ⊕⃗e∈Zn+1A(⃗e),  where A(⃗e) consists of all elements in A whose components are scalar multiples of x⃗e ∶=  xe0  0 xe1  1 ⋯xen  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' In other words, A(⃗e) is the ⃗e-isotypical summand with respect to the action  of the Gn+1  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Let us set for ⃗e ∈ Zn+1,  k(⃗e) ∶= max{i ∣ ei ≥ 0}  (which is equal to −∞ if all ei are negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' There is then a standard homotopy Q defined  on an element γ ∈ A(⃗e)p by Q(γ)i0i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip−1 = γk(⃗e)i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip−1 if k(⃗e) > −∞ and Q(γ)i0i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='ip−1 = 0  otherwise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=', if all ei are negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  For a Laurent monomial x⃗e and a subset I = {i0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',ip} ⊂ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',n} such that I ⊃ {0 ≤  i ≤ n∣ei < 0}, let us denote by x⃗e  I the element of Ap given by  (x⃗e  I)j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='jp = {x⃗e  if {j0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',jp} = I,  0  otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  TEN COMPATIBLE POISSON BRACKETS ON P5  3  Note that the condition I ⊃ {0 ≤ i ≤ n∣ei < 0} guarantees that x⃗e is a regular section of the  appropriate line bundle over Ui0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Clearly, these elements form a basis for A and our  homotopy operator Q is given by  Q(x⃗e  I) =  ⎧⎪⎪⎨⎪⎪⎩  (−1)jx⃗e  I∖k(⃗e) if k(⃗e) = ij ∈ I  0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  With these data one can in principle calculate all the higher products on the cohomology  algebra H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Below we will get explicit formulas in the case we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Calculation of m4 for P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We now specialize to the case of the projective plane  P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We will have no higher products of odd degree because H and H⊗n only live in even  degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Below we will explicitly compute the product m4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' For degree reasons it will only  be non-zero on elements e ⊗ f ⊗ g ⊗ h ∈ H⊗4 where one or two of the arguments lie in H2  and the rest in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Hence, the following special case of the multiplication in A will be  relevant: for a monomial x⃗e and a Laurent monomial x⃗e′ we have  ι0(x⃗e) ⋅ x  ⃗e′  I = x  ⃗e′  I ⋅ ι0(x⃗e) = x⃗e+⃗e′  I  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  We use the formula  m4(e,f,g,h) = −∑  T  ǫ(T)mT (e,f,g,h)  where the sum runs over all rooted binary trees with 4 leaves labeled e,f,g and h (from left  to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' For each such tree T the expressions mT(e,f,g,h) is computed by moving the  inputs through that tree, applying ι at the leaves, applying the homotopy Q on each interior  edge, multiplying elements of A at each inner vertex and finally applying the projection π  at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  We have to sum over the following five trees, which we denote T1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=',T5 respectively:  Let us first consider the case e ∈ H2 and f,g,h ∈ H0 and let’s take them all to be basis  elements of H2 and H0:  e = (xα0  0 xα1  1 xα2  2 ){0,1,2}, f = xa0  0 xa1  1 xa2  2 , g = xb0  0 xb1  1 xb2  2 , h = xc0  0 xc1  1 xc2  2  where α0,α1,α2 < 0 and ai,bi,ci ≥ 0 for i = 0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' In this case only one of the trees above  can be non-zero in the expression for m4(e,f,g,h), namely T5, because in all other trees  at some point the homotopy Q will be applied to an element of A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Below is a picture of  the different summands in A● and the possible ways the homotopy Q can map a monomial  4  VILLE NORDSTROM AND ALEXANDER POLISHCHUK  element in each summand  H2  A2 ∶  0,1,2  A1 ∶  0,1  0,2  1,2  A0 ∶  0  1  2  H0  ι2  (1)  (3)  (2)  (4)  π0  When computing mT5(e,f,g,h) we think of it as e moving through this diagram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' at every  node it gets multiplied by one of the other arguments and then it moves downwards along  one of the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We see that to be non-zero we have to go either (1) followed by (2) or (3)  followed by (4) (so that we land in ●2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We claim that only the second route is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  The reason is because at each node we multiply e by a monomial so the exponents of  x0,x1,x2 will not decrease at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' By the definition of Q, if we go along (1) we must  have that the exponent of x1 was non-negative and the exponent of x2 was negative after  the multiplication at ●0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' After performing the multiplication at ●0,2 the exponent of  x1 will still be non-negative and it follows then from the definition of Q that (2) is not  possible after (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Now comes the computation of mT5(e,f,g,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Below, we denote by µ the multiplication  in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  mT5(e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='h) =  πµ(Qµ(Qµ(e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='h) =  πµ(Qµ(Q(xα0+a0  0  xα1+a1  1  xα2+a2  2  ){0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='h)  (∗)=  πµ(Qµ((xα0+a0  0  xα1+a1  1  xα2+a2  2  ){1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='h) =  πµ(Q(xα0+a0+b0  0  xα1+a1+b1  1  xα2+a2+b2  2  ){1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='h)  (∗∗)  =  π(µ((xα0+a0+b0  0  xα1+a1+b1  1  xα2+a2+b2  2  ){2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='h)) =  π((xα0+a0+b0+c0  0  xα1+a1+b1+c1  1  xα2+a2+b2+c2  2  ){2})  (∗∗∗)  =  xα0+a0+b0+c0  0  xα1+a1+b1+c1  1  xα2+a2+b2+c2  2  TEN COMPATIBLE POISSON BRACKETS ON P5  5  where (∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' (∗∗) are (∗ ∗ ∗) means we get zero unless the following conditions hold  (∗)  ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩  α0 + a0 ≥ 0  α1 + a1 < 0  α2 + a2 < 0  (∗∗) {α1 + a1 + b1 ≥ 0  α2 + a2 + b2 < 0  (∗ ∗ ∗)  ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩  α0 + a0 + b0 + c0 ≥ 0  α1 + a1 + b1 + c1 ≥ 0  α2 + a2 + b2 + c2 ≥ 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  In the end we have  m4(e,f,g,h) = −mT5(e,f,g,h) = −ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' ⃗a,⃗b, ⃗c) ⋅ x⃗α+⃗a+⃗b+⃗c,  where  ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' ⃗a,⃗b, ⃗c) ∶=  ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩  1 if  α0 + a0 ≥ 0  α1 + a1 < 0  α1 + a1 + b1 ≥ 0  α2 + a2 + b2 < 0  α2 + a2 + b2 + c2 ≥ 0  0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Similarly we compute m4 applied to e,f,g,h in any given order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We have  m4(e,f,g,h) = − ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' ⃗a,⃗b, ⃗c) ⋅ x⃗α+⃗a+⃗b+⃗c,  m4(f,e,g,h) =[−ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' ⃗a,⃗b, ⃗c) + ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='⃗b, ⃗a, ⃗c) − ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='⃗b, ⃗c, ⃗a)] ⋅ x⃗α+⃗a+⃗b+⃗c,  m4(f,g,e,h) =[ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='⃗b, ⃗a, ⃗c) − ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='⃗b, ⃗c, ⃗a) + ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' ⃗c,⃗b, ⃗a)] ⋅ x⃗α+⃗a+⃗b+⃗c,  m4(f,g,h,e) =ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' ⃗c,⃗b, ⃗a) ⋅ x⃗α+⃗a+⃗b+⃗c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Feigin-Odesskii brackets  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Bivectors on projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' It is well known that every Gm-invariant bivector  on a vector space V leads to a bivector on the projective space PV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' A bivector on V can be  thought of as a skew-symmetric bracket {⋅,⋅} on the polynomial algebra S(V ∗), which is a  biderivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Such a bracket is Gm-invariant if and only if the bracket of two linear forms  is a quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' In other words, such a bracket can be viewed as a skew-symmetric  pairing  b ∶ V ∗ × V ∗ → S2(V ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  6  VILLE NORDSTROM AND ALEXANDER POLISHCHUK  The corresponding bivector Π on the projective space PV is determined by the skew-  symmetric forms Πv on T ∗  v PV for each point ⟨v⟩ ∈ PV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We have an identification  T ∗  v PV = ⟨v⟩∨ ⊂ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  It is easy to see that under this identification we have  Πv(s1 ∧ s2) = b(s1,s2)(v),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1)  where s1,s2 ∈ ⟨v⟩∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Here we take the value of the quadratic form b(s1 ∧ s2) at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  We can use the above formula in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Namely, suppose for some bivector Π on PV  we found a skew-symmetric pairing b such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Then the Gm-invariant bracket  {⋅,⋅} on S(V ) given by b induces the bivector Π on PV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Note that if Π is a Poisson bivector  on PV , it is not guaranteed that the Gm-invariant bracket {⋅,⋅} on S(V ) is also Poisson,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=', satisfies the Jacobi identity (but it is known that {⋅,⋅} can be chosen to be Poisson,  see [1], [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Recollections from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Let ξ be a line bundle of degree n on an elliptic curve C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  We fix a trivialization ωC ≃ OC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Then the associated Feigin-Odesskii Poisson structure  Π (to which we will refer as FO bracket) on PH1(ξ−1) ≃ PH0(ξ)∗ is given by the formula  (see [3, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1])  Πφ(s1 ∧ s2) = ⟨φ,MP(s1,φ,s2)⟩,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2)  where ⟨φ⟩ ∈ PExt1(ξ,O), and s1,s2 ∈ ⟨φ⟩⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Here we use the Serre duality pairing ⟨⋅,⋅⟩  between H0(ξ) and H1(ξ−1) and the triple Massey product  MP ∶ H0(ξ) ⊗ H1(ξ−1) ⊗ H0(ξ) → H0(ξ)  that also agrees with the triple product m3 obtained by homological perturbation from the  natural dg enhancement of the derived category of coherent sheaves on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' There is some  ambiguity in a choice of m3 but for s1,s2 ∈ ⟨φ⟩⊥, the expression in the right-hand side of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Next, assume that S is a smooth projective surface, L is a line bundle on S such that  H∗(L ⊗ KS) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Then for each smooth (connected) anticanonical divisor C ⊂ S (which is  an elliptic curve), we have a natural restriction map  H0(S,L) → H0(C,L∣C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  The exact sequence  0 → LKS  F✲ L → LC → 0  shows that under our assumptions this restriction map is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Thus, the FO bracket on PH0(L∣C)∗ associated with (C,L∣C) (defined up to rescaling)  can be viewed as a Poisson structure on a fixed projective space PV ∗, where  V ∶= H0(S,L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  By [3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='4], the Poisson brackets on PV ∗ associated with different anticanonical  divisors are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' More precisely, we get a linear map from H0(S,K−1  S ) to the space  of bivectors on PV ∗, whose image lies in the space of Poisson brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  TEN COMPATIBLE POISSON BRACKETS ON P5  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Feigin-Odesskii bracket for an anticanonical divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We keep the data (S,L)  of the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Let i ∶ C ↪ S be an anticanonical divisor in S, with the equation  F ∈ H0(S,K−1  S ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We want to write a formula for the FO bracket Π = ΠF on PV ∗ in terms  of higher products on the surface S and the equation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' For this we rewrite the right-hand  side of formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Let us write the triple product in this formula as MP C to remember  that it is defined for the derived category of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' (i) In the above situation, given e ∈ V ∗ and s1,s2 ∈ ⟨e⟩⊥, one has  ⟨e,MP C(s1∣C,e,s2∣C)⟩ = ⟨m4(F,s1,e,s2) − m4(s1,F,e,s2),e⟩,  where we use the identification V ∗ ≃ H2(S,L−1KS) given by Serre duality and consider the  A∞-products on S,  m4 ∶ H0(K−1  S )H0(L)H2(L−1KS)H0(L) → H0(L), H0(L)H0(K−1  S )H2(L−1)H0(L) → H0(L),  obtained by the homological perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  (ii) Assume that a generic anticanonical divisor is smooth (and connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Then  ΠF∣e(s1 ∧ s2) ∶= ⟨m4(F,s1,e,s2) − m4(s1,F,e,s2),e⟩,  gives a collection of compatible Poisson brackets on PV depending linearly on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' (i) By Serre duality, H∗(S,L−1) = 0, so the map  H1(C,L−1∣C) → H2(S,L−1KS),  induced by the exact sequence  0 → L−1KS → L−1 → L−1∣C → 0,  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' It is a standard fact that this isomorphism is the dual to the isomor-  phism H0(S,L) → H0(C,L∣C) given by the restriction, via Serre dualities on S and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Let  us denote by eC ∈ H1(C,L−1∣C) the element corresponding to e ∈ H2(S,L−1KS) under the  above isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  We claim that the triple Massey product MP C(s1∣C,eC,s2∣C) = m3(s1∣C,eC,s2∣C) corre-  sponding to the arrows  OC  s2∣C✲ L∣C  [1]✲ OC  s1∣C✲ L∣C  agrees with the corresponding triple Massey product on S,  OS → L  [1]✲ OC → L∣C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Indeed, the relevant spaces are identified via the restriction maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Let r ∶ OS → OC,  rL ∶ L → L∣C be the natural maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Then we have to check that for s1,s2 ∈ ⟨e⟩⊥ ⊂ H0(S,L),  one has  m3(s1∣C,eC,s2∣C)r ≡ m3(s1∣C,eCrL,s2)  mod ⟨s1∣Cr,s2∣Cr⟩,  where we view this as equality of cosets in Hom(OS,L∣C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' The A∞-identities imply that  m3(s1∣C,eC,s2∣C)r ≡ m3(s1∣C,eC,s2∣Cr) ± s1∣Cm3(eC,s2∣C,r),  where s2∣Cr = rLs2, and  m3(s1∣C,eC,rLs2) = m3(s1∣C,eCrL,s2) ± s1∣Cm3(eC,rL,s2) ± m2(s1∣C,eC,rL)s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  8  VILLE NORDSTROM AND ALEXANDER POLISHCHUK  Combining these two identities we deduce our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Thus, it is enough to calculate the Massey product MP(s1∣C,eCrL,s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Using the exact  sequences above we can represent OC (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=', LC) by the twisted complex [KS[1] → OS]  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=', [LKS[1] → L]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  In terms of these resolutions the elements of Ext1(L,OC) get represented by Ext2(L,KS) ⊂  hom●(L,[KS[1] → OS]), while the element of Hom(OC,L∣C) corresponding to s ∈ H0(S,L) ≃  H0(C,L∣C) is given by the natural map of twisted complexes induced by the multiplication  by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' The elements of Hom(OS,L∣C) are identified with Hom(OS,L) ≃ hom0(OS,[LKS[1] →  L]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Thus, the m3 product we are interested is given by the following triple product in the  category of twisted complexes over S:  OS  L  s2  ❄  KS[1]  e  ❄  F  ✲ OS  LKS[1]  s1  ❄  F  ✲ L  s1  ❄  where we view e as a morphism of degree 1 from L to KS[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Now the formula for m3 on  twisted complexes gives  m4(F,s1,e,s2) − m4(s1,F,e,s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  (ii) It is clear that ΠF gives a linear map from H0(S,ω−1  S ) to the space of bivectors on PV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  By (i), for generic F we get a Poisson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Hence, this is true for all F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' The case leading to 10 compatible brackets on P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We can apply Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1 to the case S = P2 and L = O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Note that the assumptions are satisfied in this case  since LKS = O(−1) has vanishing cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Thus, for each F ∈ H0(P2,O(3)) giving a  smooth cubic, we get a formula for the FO-bracket ΠF on PH0(P2,O(2))∗ = P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Hence,  we get a family of 10 (the dimension of H0(P2,O(3)) compatible brackets on P5 (we also  know this from [3, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' The fact that these 10 brackets are linearly independent  follows from the compatibility of this construction with the GL3-action and is explained  in [3, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  TEN COMPATIBLE POISSON BRACKETS ON P5  9  Now we will derive formulas for the the brackets {,}F on the algebra of polynomials in  6 variables which induce the above Poisson brackets on PV ≃ P5, where  V = H0(P2,O(2))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  They depend linearly on F, so we will just give formulas for {,}x⃗c, where x⃗c runs through  all 10 monomials of degree 3 in (x0,x1,x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Let us set  ∆(n) ∶= {{(a0,a1,a2) ∈ Z3 ∣a0 + a1 + a2 = n,ai ≥ 0 for i = 0,1,2} if n ≥ 0  {(α0,α1,α2) ∈ Z3 ∣α0 + α1 + α2 = n,αi < 0 for i = 0,1,2} if n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Note that {x⃗e ∣ ⃗e ∈ ∆(n)} forms a basis for H0(P2,O(n)) when n ≥ 0, while {x⃗e  {0,1,2} ∣ ⃗e ∈  ∆(n)} is a basis for H2(P2,O(n)) when n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' In particular, we use {x⃗a ∣ ⃗a ∈ ∆(2)} as  a basis in V ∗ = H0(P2,O(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Our brackets should associate to a pair of elements of this  basis a quadratic form in the same variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' One has for ⃗a,⃗b ∈ ∆(2), ⃗c ∈ ∆(3),  {x⃗a,x  ⃗b}x⃗c ∶=  ∑  ⃗a′,⃗b′∈∆(2)  [∑  σ  −sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗a′, ⃗b′)]x  ⃗a′x  ⃗b′  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='3)  where the second sum is over the symmetric group on the letters {a,b,c} and  ˜ρ(⃗a,⃗b, ⃗c, ⃗a′, ⃗b′) ∶=  ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩  1 if  a′  0 ≤ a0 − 1  a′  1 > a1 − 1  a′  1 ≤ a1 + b1 − 1  a2 + b2 < a′  2 + 1  c2 + a2 + b2 ≥ a′  2 + 1  a′  0 + b′  0 = a0 + b0 + c0 − 1  a′  1 + b′  1 = a1 + b1 + c1 − 1  0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' By Serre duality, we can identify V = H0(P2,O(2))∗ with H2(P2,O(−5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  By  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1, the bracket {x⃗a,x⃗b}x⃗c is the quadratic form on V ≃ H2(P2,O(−5)) given  by  Q(e) ∶= ⟨e,m4(x⃗c,x⃗a,e,x  ⃗b) − m4(x⃗a,x⃗c,e,x  ⃗b)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  We can write  e =  ∑  ⃗α∈∆(−5)  c⃗αx⃗α  {0,1,2} ∈ H2(P2,O(−5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Using the formulas for m4 from the end of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2 we get  Q(e) =  ∑  ⃗α, ⃗β∈∆(−5)  [∑  σ  −sgn(σ)ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='σ⃗a,σ⃗b,σ⃗c)]δ(⃗α, ⃗β, ⃗a,⃗b, ⃗c)c⃗αc ⃗β,  10  VILLE NORDSTROM AND ALEXANDER POLISHCHUK  where the second sum runs over the symmetric group on the letters {a,b,c} and  δ(⃗α, ⃗β, ⃗a,⃗b, ⃗c) = {1 if ⃗α + ⃗β + ⃗a + ⃗b + ⃗c = (−1,−1,−1)  0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  We have to show that the element in S2(H0(P2,O(2))) given by the right-hand side of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='3) defines the same quadratic form Q on H2(P2,O(−5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' To see this we apply it to the  element e = ∑⃗α∈∆(−5) c⃗αx⃗α  {0,1,2} ∈ H2(P2,O(−5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' For ⃗α ∈ O(−5) we set ⃗α∗ ∶= (−1,−1,−1)− ⃗α  and then we compute  (  ∑  ⃗a′,⃗b′∈∆(2)  [∑  σ  −sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗a′, ⃗b′)]x  ⃗a′x  ⃗b′)(e) =  ∑  ⃗α, ⃗β∈∆(−5)  ∑  ⃗a′,⃗b′∈∆(2)  [∑  σ  −sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗a′, ⃗b′)]⟨x  ⃗a′,x⃗α  {0,1,2}⟩⟨x  ⃗b′,x  ⃗β  {0,1,2}⟩c⃗αc ⃗β =  ∑  ⃗α, ⃗β∈∆(−5)  [∑  σ  −sgn(σ)˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗α∗, ⃗β∗)]c⃗αc ⃗β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Now it only remains to note that ˜ρ(σ⃗a,σ⃗b,σ⃗c, ⃗α∗, ⃗β∗) = ρ(⃗α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='σ⃗a,σ⃗b,σ⃗c)δ(⃗α, ⃗β, ⃗a,⃗b, ⃗c) for  any permutation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  □  Remarks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Note that when we take ⃗c = (0,0,3) only two permutations σ, namely,  σ = 1 and σ = (a b), can give non-zero terms in the formula of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  When  ⃗c = (1,2,0) all permutations except σ = 1 and σ = (a b) may give non-zero terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' When  ⃗c = (1,1,1) all permutations can give non-zero terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' We do not claim that formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='3) define Poisson brackets and are compatible on the  algebra of polynomials in 6 variables, only that this holds for the induced brackets on P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
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+page_content=' Odesskii, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Wolf, Compatible quadratic Poisson brackets related to a family of elliptic curves,  arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='1299  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Polishchuk, Algebraic geometry of Poisson brackets, Journal of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Sciences 84 (1997) 1413–1445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' Polishchuk, Poisson structures and birational morphisms associated with bundles on elliptic curves,  IMRN 13 (1998), 683–703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='  Department of Mathematics, University of Oregon, Eugene, OR 97403, USA  Email address: villen@uoregon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='edu  TEN COMPATIBLE POISSON BRACKETS ON P5  11  Department of Mathematics, University of Oregon, Eugene, OR 97403, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content=' National  Research University Higher School of Economics  Email address: apolish@uoregon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FQT4oBgHgl3EQf0jbn/content/2301.13417v1.pdf'}

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+Characterizing Quantile-varying Covariate
+Effects under the Accelerated Failure Time
+Model
+Harrison T. Reeder
+Massachusetts General Hospital Biostatistics
+Department of Medicine, Harvard Medical School
+Kyu Ha Lee
+Departments of Nutrition, Biostatistics, and Epidemiology,
+Harvard T.H. Chan School of Public Health
+Sebastien Haneuse
+Department of Biostatistics, Harvard T.H. Chan School of Public Health
+Abstract
+An important task in survival analysis is choosing a structure for the relationship
+between covariates of interest and the time-to-event outcome. For example, the accel-
+erated failure time (AFT) model structures each covariate effect as a constant multi-
+plicative shift in the outcome distribution across all survival quantiles. Though parsi-
+monious, this structure cannot detect or capture effects that differ across quantiles of
+the distribution, a limitation that is analogous to only permitting proportional hazards
+in the Cox model. To address this, we propose a general framework for quantile-varying
+multiplicative effects under the AFT model. Specifically, we embed flexible regression
+structures within the AFT model, and derive a novel formula for interpretable effects
+on the quantile scale. A regression standardization scheme based on the g-formula is
+proposed to enable estimation of both covariate-conditional and marginal effects for an
+exposure of interest. We implement a user-friendly Bayesian approach for estimation
+and quantification of uncertainty, while accounting for left truncation and complex cen-
+soring. We emphasize the intuitive interpretation of this model through numerical and
+graphical tools, and illustrate its performance by application to a study of Alzheimer’s
+disease and dementia.
+Keywords: Accelerated failure time model; Bayesian survival analysis; Left-truncation; Time-
+varying coefficients; Time-varying covariates
+This is the pre-peer reviewed, “submitted” version of the following article which is published in Biostatistics by Oxford University Press:
+Reeder HT, Lee KH, Haneuse S. Characterizing quantile-varying covariate effects under the accelerated failure time model. Biostatistics. kxac052.
+2022 Jan 04. doi: 10.1093/biostatistics/kxac052. PMID: 36610077.
+Arxiv will be updated with the final peer-reviewed “accepted” version of the manuscript after a 24 month embargo period.
+1
+arXiv:2301.03057v1  [stat.ME]  8 Jan 2023
+
+1
+Introduction
+Modeling the relationship between a time-to-event outcome T and a vector of covariates X
+requires choosing a structure for the covariate effects. The proportional hazards model is
+by far the most commonly used model, specifying a constant multiplicative effect on the
+hazard of the outcome, yielding a ‘hazard ratio.’ Though ubiquitous, hazard ratios can be
+difficult to interpret, and the constant effect across time—that is, the ‘proportionality’ of
+the hazards—is not always plausible (Hern´an, 2010; Uno et al., 2015). As an alternative, the
+accelerated failure time (AFT) model directly describes shifts in the outcome distribution
+between populations having different characteristics, via multiplicative effects on event time
+quantiles (Wei, 1992). Specifically, every survival quantile is multiplied by a constant ‘accel-
+eration factor,’ equivalent to a horizontal stretching or compressing of the survivor function.
+In other words, the times by which 10 percent of events occur, or 90 percent, or 50 percent
+(i.e., the median survival time), or any other quantile, are shifted by the same multiplicative
+(or relative) constant. This common effect across quantiles is the central feature of the AFT
+model, making it highly interpretable because contrasts of survival quantiles are tangible
+and often clinically meaningful. Despite the parsimony of a constant multiplicative effect,
+in some settings it may be important to allow for more flexible effects across quantiles. For
+example, consider the study of Alzheimer’s disease (AD) and dementia among older adults.
+Prospective cohort studies of incident AD and dementia typically enroll subjects and follow
+them over decades, often subject to left truncation and sometimes complex censoring. Age
+at AD onset among those with a particular risk factor, for example, may skew earlier than
+among those without the risk factor. However, because AD is a complex disease that can
+arise over a long time scale, baseline risk factors may not affect the entire distribution uni-
+formly. This could occur if, for example, a risk factor did not affect the timing of ‘early-onset’
+cases, but made ‘late-onset’ cases occur sooner.
+Modeling hazard ratios flexibly across time is a well-known and commonly used tool un-
+der the Cox model, but analogous extensions of the AFT model are not well-studied. Very
+recently, a paper by Crowther et al. (2022) suggests a frequentist spline-based AFT model
+and discusses potential for time-varying effects. However, their work considers a less common
+interpretation of the acceleration factor on the scale of log time, rather than investigating the
+potential for flexibility on the quantile scale. Moreover, their paper does not present any nu-
+merical results for the use of flexible effects. Separately, a recent paper by Pang et al. (2021)
+considers a Frequentist spline-based AFT model using a completely different formulation
+derived from Prentice and Kalbfleisch (1979), requiring a specialized estimation algorithm
+and bootstrapping for inference. However, these papers do not incorporate left truncation
+or complex censoring, or consider effects of time-dependent covariates that commonly arise
+in longitudinal studies, for which the resulting relationship varies both over the trajectory
+of the covariate, and over the survival quantiles.
+2
+
+In this paper, we extend the AFT model to allow flexible acceleration factors that vary
+across quantiles, while simultaneously accommodating left-truncation, complex censoring,
+and time-varying covariates. Our approach builds on a time-varying AFT model first in-
+troduced in Cox and Oakes (1984) but seemingly largely overlooked in the literature, and a
+general framework for flexible covariate effect specification. We illustrate how AFT regres-
+sion coefficients specified to vary over time can be inverted into quantile-varying acceleration
+factors, and we develop a regression standardization scheme based on the g-formula to allow
+estimation of both covariate-conditional and marginal acceleration factors for an exposure of
+interest. We propose a Bayesian estimation approach for this modeling framework using the
+Stan language, which allows rigorous quantification of uncertainty and increased modeling
+flexibility. Through this investigation, we also uncover new insights into the use of binary
+time-varying covariates under the AFT model, and present novel tools for modeling and
+visualizing such effects. This further expands the AFT modeling toolkit to cover many ex-
+tensions commonly used under the Cox model. We motivate these methods with an in-depth
+analysis of the Religious Orders Study and Memory and Aging Project prospective cohort
+studies of AD and dementia (Bennett et al., 2018).
+2
+The Accelerated Failure Time Model
+The standard AFT model with time-invariant effects can be written as a log-linear model of
+time:
+log(T) = X
+Tβ + ϵ,
+where ϵ is a random error term and β is a vector of regression coefficients corresponding
+with covariates X. We denote the exponentiated error T0 = exp(ϵ), which represents a hy-
+pothetical random variable drawn from the “baseline distribution” having survivor function
+S0. It is straightforward to show that this model structures covariate effects such that the
+distribution of event times among subjects having covariate pattern x, denoted Tx, is directly
+shifted from the baseline distribution by the transformation
+Tx × exp(−x
+Tβ) ∼ S0.
+Based on this connection, an equivalent representation of the standard AFT model is given
+directly via the baseline survivor function S0 as
+S(t | X) = S0(t × exp(−X
+Tβ)).
+(1)
+The AFT model admits a direct interpretation of covariate effects as multiplicative shifts
+of the survival quantiles. For any particular quantile p, define t(p)
+x
+and t(p)
+0
+to be the pth
+quantile times under x and baseline respectively. Then
+p = S(t(p)
+x | x) = S0(t(p)
+x × exp(−x
+Tβ)) = S0(t(p)
+0 ).
+3
+
+Solving for the pth quantile survival time under x yields
+t(p)
+x = S−1(p | x) = S−1
+0 (p) exp(x
+Tβ).
+The acceleration factor between two arbitrary covariate patterns x and x′ is then defined as
+the ratio of pth quantiles,
+t(p)
+x /t(p)
+x′ =
+S−1
+0 (p) exp(xTβ)
+S−1
+0 (p) exp((x′)Tβ) = exp((x − x′)
+Tβ).
+Under the standard AFT model, the acceleration factor does not depend on the form of S0
+or the value of p, and thus characterizes a constant multiplicative covariate effect across the
+entire distribution.
+2.1
+AFT model with time-varying components
+In the standard AFT model (2), the covariate-adjusted survivor function is directly charac-
+terized by the time shift defined by t × exp(−XTβ). Towards a more flexible AFT model,
+we replace this time shift with a general increasing function V (t | X), yielding the covariate-
+adjusted survivor function
+S(t | X) = S0 (V (t | X)) .
+(2)
+This formulation, first discussed by Cox and Oakes (1984) in the context of time-varying
+covariates, reduces to the standard AFT when V (t | X) = t × exp(−XTβ), while also admit-
+ting other temporal specifications of the relationship between covariates and the outcome
+distribution. In fact, one interpretation of this V function is as a transformation linking the
+distribution of Tx under covariates x, and the baseline distribution of T0,
+V (Tx | x) ∼ S0.
+Under this extended AFT model (2.1), the pth quantile survival time for subjects under
+covariate pattern x is
+t(p)
+x = S−1(p | x) = V −1(S−1
+0 (p) | x).
+Now it may no longer be the case that the ratio of pth quantile survival times between
+covariate patterns x and x′ is a constant factor. Instead, the more general quantile-varying
+acceleration factor is
+ξ(p | x, x′, S0) = t(p)
+x /t(p)
+x′ = S−1(p | x)
+S−1(p | x′) = V −1(S−1
+0 (p) | x)
+V −1(S−1
+0 (p) | x′),
+(3)
+with notation explicitly capturing the additional potential for dependence on p and S0.
+4
+
+2.1.1
+Examples and Interpretation
+To emphasize both the flexibility and interpretability of this new quantity, Figure 1 shows
+sample survivor curves and corresponding acceleration factors under simple forms of quantile-
+varying effect for a single contrast between exposure levels X = 1 and X = 0, with baseline
+S0(t) = exp(−0.3t). For simplicity we will interpret the effects at p = 0.75 and p = 0.25,
+which represent the time by which 25% and 75% of people experience the event, respectively.
+As a reference point, the blue curve (second row of the legend) in each figure shows a
+constant acceleration factor of exp(0.5) ≈ 1.65, constant across quantiles. The green curve
+(fourth row of the legend) shows a protective effect that is increasingly pronounced among
+later-onset cases, with ξ(0.75 | 1, 0) = 1.25 and ξ(0.25 | 1, 0) = 2. In words, the estimated
+time by which 25% of the exposed die is 1.25 times as great as that among the unexposed,
+but the estimated time by which 75% of the exposed die is 2 times greater than unexposed.
+Conceptually, this form of protective effect corresponds with delayed onset of all cases among
+the exposed, but specifically a much longer tail of late-onset cases compared to a standard
+AFT protective effect.
+The orange curve (third row of the legend) shows a more nuanced effect that delays the
+earliest cases, while also accelerating later onset cases. Numerically, ξ(0.75 | 1, 0) = 1.65
+and ξ(0.25 | 1, 0) = 0.9, meaning the estimated time by which 25% of the exposed die is
+1.65 times as great as that among the unexposed, but the estimated time by which 75% of
+the exposed die is only 0.9 times as great as among the unexposed. Conceptually, this form
+of effect is a ‘compressing’ of the outcome distribution, with earlier events being delayed
+and later events being accelerated. This is visible in the relative steepness of the survivor
+curve, with more than 50% of all events occurring between times 2 and 4. Furthermore,
+this represents an effect with ‘crossing survivor curves’, which despite being common in
+certain health research domains cannot be modeled by standard proportional hazards or
+AFT models. In summary, we see that this approach to conceptualizing covariate effects for
+time-to-event outcomes yields nuanced and interpretable insights beyond what is available
+from standard proportional hazards or AFT models.
+3
+Model Definition
+The proposed quantile-varying AFT model is purposefully general with respect to the base-
+line survivor distribution S0 and the time-varying covariate process V . In this section we
+outline several choices for specifying these model components, weighing tradeoffs between
+flexibility, stability, and computation. While this modeling framework in principle admits
+estimation under both frequentist and Bayesian paradigms, we focus on the latter approach
+and employ a Markov Chain Monte Carlo (MCMC) estimation routine via the No-U-Turn
+sampler implemented in the Stan language (Carpenter et al., 2017).
+5
+
+3.1
+Specification of the covariate process V
+For ease of exposition, we will consider a d length vector of baseline covariates X, of which
+an exposure of interest X1 is specified with a flexible regression effect. However, this can
+easily be expanded to allow multiple such exposures of interest.
+The form of the covariate process V dictates the potential shapes the quantile-varying
+acceleration factor for X1 can take, and requires a balance of flexibility and stability. We
+focus on spline-based methods, which require a vector of knots τ characterizing a set of J
+basis functions B1, . . . , BJ, and corresponding coefficients α = (α1, . . . , αJ)T. This results in
+the specification
+V (t | X) = t × exp
+�
+−X
+Tβ − X1
+J
+�
+j=1
+αjBj(t | τ)
+�
+,
+(4)
+Note that when α = 0, then this reduces to the standard AFT model, allowing straight-
+forward model comparison to assess the flexible effect specification. Furthermore, letting
+B′
+j(t | τ) = dBj(t | τ)/dt, then the derivative of the covariate process, which is used in
+likelihood computation, has the simple form
+v(t | X) = d
+dtV (t | τ) = V (t | τ)
+�
+1
+t − X1
+J
+�
+j=1
+αjB′
+j(t | τ)
+�
+.
+One specification inspired by the parametric proportional hazards spline model of Roys-
+ton and Parmar (2002) and discussed by Crowther et al. (2022) is the natural cubic spline
+basis, which combines cubic polynomial basis functions with a restriction that the ends be-
+yond the lower and upper boundary knots be linear.
+Numerically stable forms for each
+natural cubic spline basis function Bk and B′
+k are readily available in statistical software,
+and the resulting V combines flexibility and stability, with the added advantage of being
+a smooth function of time. However, the inverse V −1(t | X) used in the quantile-varying
+acceleration factor (2.1) does not have a closed form, and must be computed numerically.
+A computationally simpler alternative is to specify V as a piecewise linear function, which
+yields a simplified analytical form and closed form inverse. Define J + 2 knots 0 = τ0 < τ1 <
+· · · < τJ < τJ+1 = ∞, with basis functions defined Bj(t | τ) = t−1(min{t, τj+1} − τj)+ where
+(z)+ = min{0, z}. Then the final specification for V simplifies to
+V (t | X) = t × exp (−X
+Tβ)
+� J
+�
+j=1
+exp (−X1αj) Bj(t | τ)
+�
+,
+with the straightforward derivative v(t | X) = exp
+�
+−XTβ − �J
+j=1 X1αjI(τj ≤ t < τj+1)
+�
+.
+Computation of the inverse is also straightward, and left to Appendix B of the Supplementary
+Materials. As above, this reduces to the standard AFT model when α = 0.
+6
+
+3.2
+Specification of the baseline distribution S0
+As with the specification of V , there are numerous possible choices of baseline distribution
+characterizing S0, both fully parametric and semi-parametric. Parametric specifications have
+several advantages in this setting: they are computationally efficient, well-defined across all
+quantiles, have tractible inverse survivor functions, and can lead to improved efficiency in
+smaller samples. Two such parametric specifications are the log-Normal baseline distribution
+with survivor function defined by S0(t | µ, σ) = 1 − Φ(log t − µ)/σ2 where Φ(·) is the
+standard normal distribution function, and the Weibull baseline distribution defined by
+S0(t | µ, σ) = exp {[t × exp(−µ)]σ}. Let φ = (µ, σ)T denote the collection of parameters
+corresponding to the baseline distribution.
+Nevertheless, an important benefit of the Bayesian paradigm is the well-established liter-
+ature on semi-parametric AFT survival models with flexible baseline distributions, such as
+Dirichlet process mixture (DPM) models (Lee et al., 2017) and Polya tree priors (Hanson
+et al., 2009). Here we propose a transformed Bernstein polynomial (TBP) prior for S0 fol-
+lowing (Zhou and Hanson, 2018), which defines a parametric centering distribution having
+survivor function S∗
+0(t | φ) (such as the Weibull or log-Normal defined above), then applies
+a transformation using Bernstein polynomial functions to can flexibly capture a wide array
+of distributions. Formally, define the Beta(a, b) distribution function
+G(p | a, b) = Γ(a + b)
+Γ(a)Γ(b)pa−1(1 − p)b−1,
+0 ≤ x ≤ 1,
+and a vector w of length K such that �K
+k=1 wk = 1. Then the baseline survivor function is
+the linear combination
+S0(t | φ, w) =
+K
+�
+k=1
+wkG(S∗
+0(t | φ) | k, K − k + 1).
+Because the domain of G and the range of S∗
+0 are both [0,1], this represents a flexible spline
+transformation of the centering parametric distribution on the scale of survival quantiles. In
+particular, if w = (J−1, J−1, . . . , J−1)T, then S0 = S∗
+0, so the TBP specification contains the
+centering parametric model, but can also characterize a wide array of survival distribution
+shapes. An illustration is provided in Appendix D of the Supplementary Materials. To
+complete the Bayesian specification, we place a Dirichlet(θ) prior on w with θ > 0, where
+larger values of θ correspond to tighter concentration of the elements of w around J−1 and
+therefore tighter concentration of S0 around S∗
+0.
+This specification offers several advantages over other flexible baseline specifications men-
+tioned previously.
+Importantly, each G function can be computed recursively, so overall
+computation of S0 is efficient. Moreover, the TBP prior can be straightforwardly sampled
+using the No-U-Turn algorithm implemented in the Stan language, as described below. By
+contrast, many other Bayesian non-parametric specifications such as Polya trees and DPM
+7
+
+models require specialized computational methods such as custom MCMC samplers and data
+augmentation (Hanson et al., 2009; Lee et al., 2017). The main tradeoff with any flexible
+form for S0 compared to a fully parametric specification is the increased computational cost,
+both for the sampler as well as the numerical computation of the inverse function S−1
+0
+and
+associated acceleration factors.
+3.3
+Likelihood
+Another important benefit of the Bayesian approach is the ability to seamlessly handle
+arbitrary censoring and left truncation. Let (Y l, Y u) the left and right observed endpoints
+of a censoring interval around a true event time T, such that Y l ≤ T ≤ Y u. Right-censoring
+simply corresponds with Y u = ∞.
+Define the binary indicator ∆ = I(Y l = Y u) to be
+a subject observed to experience the event exactly at time Y l. Finally, let L represent the
+possible left-truncation time. Along with the baseline covariates X, denote the corresponding
+observed data for the ith subject Di = {yl
+i, yu
+i , δi, li, xi}.
+After specifying V and S0, let ψ = (β
+T, αT, φ
+T, wT)T denote the full set of parameters.
+Then assuming that censoring is non-informative of the outcome, the resulting likelihood
+contribution for subject i is then
+Li(ψ | Di) = [f0(V (yl
+i | xi))v(yl
+i | xi)]δi[S0(V (yl
+i | xi)) − S0(V (yu
+i | xi))](1−δi)
+S0(V (li | xi))
+where f0 is the density function corresponding to the baseline distribution. By convention,
+S0(∞) = 0, so under right-censoring this reduces to the standard censored data likelihood.
+3.4
+Bayesian Computation and Prior Specification
+To implement this modeling framework, we propose Bayesian estimation via the No-U-Turn
+sampler implemented by the Stan language (Carpenter et al., 2017). In brief, this MCMC
+algorithm uses gradient information on the log-posterior to generate Markov transitions that
+efficiently explore the posterior distribution. This choice reflects our goal of developing a
+practical and accessible methodology, as our implementation can be easily called from R via
+the rstan package with minimal algorithmic tuning (Stan Development Team, 2020).
+To complete our model specification, we consider priors on the parameters β, α, and φ.
+The No-U-Turn sampler does not require or leverage conjugacy between prior and posterior,
+so prior distributions can be chosen or adjusted without changing the implementation of
+the sampler.
+In the application below, we adopt flat priors for regression parameters β
+and α. For the parametric (centering) distribution, we also adopt a flat prior for the log
+location parameter log µ, and for the scale parameter a σ ∼ Gamma(aσ, bσ) prior.
+The
+TBP prior is defined by a w ∼ Dirichlet(θ) prior for the weights, and we adopt a θ ∼
+8
+
+Gamma(aθ, bθ) hyperprior on θ, regulating the level of flexibility around the parametric
+centering distribution.
+3.5
+Model Evaluation and Comparison
+A conceptual benefit of our proposed modeling framework is that the flexible structures
+naturally encompass simpler models: the standard AFT model is nested within the flexible
+effect specification of covariate process V , and a fully parametric baseline is nested within the
+TBP prior for S0. In this section, we propose a model evaluation metric to inform decisions
+regarding the necessary level of model complexity, facilitated by the Stan language and the
+loo package in R (Vehtari et al., 2017).
+The expected log pointwise predictive density (ELPD) is a metric that evaluates how well
+a fitted model can predict future out-of-sample data, with larger values indicating better
+predictive ability. For n future observations �y1, . . . , �yn, the ELPD is defined via the posterior
+predictive density p(�y | D) as
+ELPD =
+n
+�
+i=1
+�
+log p(�yi | D)d�yi.
+While typically future out-of-sample data is not available, the ELPD can be estimated by
+leave-one-out cross validation by averaging the log posterior predictive distribution for each
+observed data point of a model fit excluding that data point. This quantity can in turn
+be estimated efficiently from a single Bayesian model fit via Pareto smoothed importance
+sampling, which we denote �
+ELPDpsis-loo (Vehtari et al., 2017), and has been shown to exhibit
+improved performance relative to other common Bayesian model criteria, such as Deviance
+Information Criterion (DIC).
+3.6
+Computation of Regression Standardized Acceleration Factors
+Importantly, under the covariate process V defined by (3.1), the quantile-varying accelera-
+tion factor (2.1) depends on the specified values of all covariates x and x′, not just those
+that differ. This conditionality on the values of all covariates may be insightful if interest
+is in assessing effect heterogeneity in particular subpopulations defined by specific covari-
+ate patterns. However, practical interest is often in assessing the effect of an exposure in
+a population standardized with respect to the other covariates. Therefore, in this section
+we propose a regression standardization approach to estimating the quantile-varying accel-
+eration factor for a particular covariate of interest, averaged over the distribution of other
+covariates. Conceptually, the goal is to first estimate the survivor curves we would observe
+in the population if everyone was alternately exposed or unexposed, and then back out the
+quantile-varying acceleration factor that relates the two curves.
+9
+
+For clarity, consider a single binary exposure of interest X, and vector of additional
+covariates Z. Then the marginal ratio of interest is
+ξ(p | X = 1, X′ = 0) = S−1(p | X = 1)
+S−1(p | X = 0).
+Following Rothman et al. (2021) and Sj¨olander (2016), define the survivor function for
+X = x, standardized to the distribution of Z, as
+SZ(t | x) = EZ[P(T > t | X = x, Z)].
+Using standardized survivor functions, we define the standardized quantile-varying acceler-
+ation factor as
+ξZ(p | X = 1, X′ = 0) = [SZ]−1(p | X = 1)
+[SZ]−1(p | X = 0).
+where [SZ]−1(p | X = x) is the function solving SZ(t | X = x) − p = 0 for t.
+This
+contrast represents the magnitude of the horizontal shift in the standardized survivor curve
+SZ between X = 1 and X = 0, at each quantile p.
+To estimate and quantify uncertainty for these contrasts, we develop a novel approach
+based on the Bayesian g-formula (Keil et al., 2018). In brief, for each MCMC draw m =
+1, . . . , M, for each X = x we compute the standardized survivor function
+S(m)
+Z (t | X = x) = n−1
+n
+�
+i=1
+S(t | X = x, Z = zi; ψ(m)),
+and then form contrast of interest
+ξ(m)
+Z (p | X = 1, X′ = 0) = [S(m)
+Z ]−1(p | X = 1)
+[S(m)
+Z ]−1(p | X = 0)
+.
+This may require numerical evaluation of the inverse standardized survivor functions. Esti-
+mating the posterior mean and credible intervals of ξZ proceeds using the mean and suitable
+quantiles of ξ(1)
+Z , . . . , ξ(M)
+Z
+.
+4
+Application: Cohort Study of Incident AD and De-
+mentia
+Motivating the proposed AFT model is the study of adverse cognitive outcomes among older
+adults, for which long timescales and complex disease etiology naturally lend themselves to
+consideration of flexible covariate effects on the quantile scale. In this section we investigate
+10
+
+risk factors for AD and dementia in older adults using data collected by the Religious Orders
+Study and Memory and Aging Project (ROSMAP) prospective cohort studies ongoing since
+1994 and 1997 respectively (Bennett et al., 2018). Our analysis focuses on flexible estima-
+tion of the association of the genetic marker APOE-ϵ4 with the timing of AD or dementia
+onset. Previous analyses of similar cohorts have simply compared incidence rates within
+age categories to examine whether this marker had differential effects through time (Kukull
+et al., 2002). So, estimating a quantile-varying acceleration factor for APOE-ϵ4 is of clinical
+relevance, while also accounting for other risk factors.
+2694 subjects were enrolled without AD or dementia between ages 65 and 86, and followed
+until withdrawal or death. Subjects underwent cognitive screening annually to diagnose onset
+of AD or dementia, and death status was monitored continuously. Table 1 summarizes a
+set of baseline binary risk factors collected on the subjects: marital status at baseline, sex,
+education level, race/ethnicity, and presence of the APOE-ϵ4 genetic variant.
+The final
+analysis dataset includes 2335 subjects with complete baseline information. The outcome
+is defined by the time of diagnosis of AD or dementia, with death treated as a censoring
+mechanism, yielding a cause-specific analysis. Because we only include subjects with age at
+least 65, the time scale of analysis is “years since age 65.” Importantly, our analysis accounts
+for the presence of left truncation (or “delayed entry”) by subjects who enroll after age 65.
+Though this framework admits interval censoring, given the short visit intervals relative to
+the timescale, for this analysis we defined the timing of AD onset at the midpoint of the
+corresponding visit interval.
+We compare the fits of standard AFT models with those having piecewise and spline
+forms for V , under Weibull and log-Normal baseline specifications as well as a TBP prior
+baseline with K = 5, centering around the Weibull distribution. We set 4 break points for
+the piecewise linear effect at 7.5 year intervals across the follow up period, and for the spline
+effect we set 2 internal knots at quantiles on the log scale. The difference between these
+specifications is due to the spline being naturally more flexible, allowing it to smooth across
+knots with irregular spacing, while the piecewise linear model requires break points that span
+the entire timespan in order to achieve flexibility. For the scale parameter we set the prior
+σ ∼ Gamma(0.3, 0.05), having prior median 1.46 and 95% central mass between 6e-5 and
+38. Finally, we fit a standard Frequentist Cox proportional hazards model for comparison.
+For the TBP concentration parameter we set a hyperprior of θ ∼ Gamma(1, 1). For each
+model we ran three chains each for 2000 adaptation iterations and 10000 samples, totalling
+30000 samples. After sampling, all potential scale reduction factors were below 1.01 and
+trace plots indicated good mixing.
+Table 3 reports the estimates of regression parameters across all AFT specifications, as
+well as frequentist results from a Cox proportional hazards model. For the AFT models,
+positive estimates of β correspond with delayed onset of AD or dementia, as do negative
+estimates for the Cox model. The coefficients estimated for white race/ethnicity, marital
+11
+
+status, female sex, and education are stable across all model specifications. Interpreting the
+Weibull AFT with constant effect of APOE-ϵ4, for example, indicates that being married is
+associated with a exp(0.09) = 1.09 times greater median time to onset of AD or dementia,
+with 95% credible interval of (1.02,1.19). Flexible effect coefficients of APOE-ϵ4 cannot be
+directly interpreted on the quantile scale, therefore we present graphical tools below.
+The top panel of Table 2 compares estimates of ELPD model criterion for each AFT
+model. In each case, the spline and piecewise-linear effect specifications outperformed the
+standard AFT specification. The log-Normal models uniformly underperformed, while the
+Weibull and TBP models performed comparably. To graphically assess the effect of APOE-ϵ4
+we report the TBP model, and present results for other specifications in Appendix A of the
+Supplementary Materials. Results were qualitatively similar for all baseline distributions,
+with the largest differences in acceleration factor only occurring in the lowest quantiles
+extrapolated beyond the observed data.
+Figure 2 shows the estimated survivor functions and corresponding quantile-varying ac-
+celeration factors for the APOE-ϵ4 genetic variant, after regression standardization over
+the distribution of the other baseline covariates. These figures confirm other findings that
+APOE-ϵ4 is associated with earlier onset of AD and dementia. However, quantile-varying
+effects also indicate that the acceleration is strongest among the earliest cases and subse-
+quently diminishes. Both piecewise and spline models estimate that the time by which the
+first 10% of those living with APOE-ϵ4 develop AD or dementia is earlier than those with-
+out the variant by a factor of about 0.5; the median times by which people develop AD
+or dementia differ by a factor of about 0.75, and the times by which 75% develop AD or
+dementia differ by a factor of about 0.85. Due to censoring of those with advanced age, the
+acceleration factor at lower quantiles reflects parametric extrapolation beyond the observed
+distribution, represented in the figure by grey shading. Nevertheless, this finding has clear
+clinical significance, indicating the particular need to monitor for early onset AD at younger
+ages among those with APOE-ϵ4.
+5
+Effects of Time-varying Covariates on the Quantile
+Scale
+In this section, we extend the proposed AFT model to incorporate binary time-varying covari-
+ates, and provide intuition and graphical tools for effectively interpreting and communicating
+corresponding effects on the quantile scale.
+To focus on intuition, consider a single time-varying covariate denoted X1(t) with con-
+stant regression effect β1. In particular, let X1(t) be a binary-valued step function, such
+as an indicator for whether a non-terminal event has occurred by time t. Formally, define
+X1(t) = I(t > tX), where tX is the time at which X1 changes. To simplify notation, consider
+12
+
+a single additional covariate time-invariant covariate X2, though inclusion of multiple addi-
+tional covariates is straightforward. Embedding these covariates directly in the structure for
+V given by (3.1) and setting α = 0 to denote a constant effect yields
+V (t | X(t)) = t × exp (−X1(t)β1 − X2β2)
+= exp (−X2β2) [min{t, tX} + (t − tX)+ exp (−β1)] .
+(5)
+With complete derivation given in Appendix C of the Supplementary Materials, the accel-
+eration factor at quantile p between two subjects depends on each person’s value of X2, the
+change time tX for X1, and the baseline distribution S0. In particular, for those with X2 = x2,
+the acceleration factor at quantile p for experiencing X1 at tX versus not experiencing X1 is
+tX
+S−1
+0 (p) exp(x2β2) + exp(β1)
+�
+1 −
+tX
+S−1
+0 (p) exp(x2β2)
+�
+.
+(6)
+This is a weighted average between 1 and exp(β1), with weight inversely proportional to the
+duration from tX to the pth quantile survival time in the comparison group, S−1
+0 (p) exp(x2β2).
+Intuitively, before tX there is no difference between the individuals, so the acceleration factor
+is 1, and then after tX the effect of X1 starts accumulating, and the acceleration factor
+gradually shifts towards exp(β1), becoming more pronounced as p extends towards 0. This
+dynamic is illustrated by example in Figure 3 below.
+Finally, a flexible effect for X1(t) can also be specified by adapting the form of (5),
+yielding
+V (t | X(t)) = e−X2β2
+�
+min{t, tX} + (t − tX)+ exp
+�
+−β1 −
+K
+�
+k=1
+αkBk(t − tX | τ)
+��
+.
+Following Haneuse et al. (2008), this specification characterizes flexibility in the effect of
+X1 over the time scale t − tX denoting time since the non-terminal event, rather than on
+the overall time scale of t, enabling evaluation of the temporal effect of X1 on its own
+timescale. Practically, this means that basis functions and knots τ must be specified on the
+corresponding time scale.
+5.1
+Effect of Incident AD and Dementia on Mortality
+To illustrate the use of the AFT framework with a time-varying binary covariate, we per-
+form a secondary analysis of the cohort study to evaluate the association between onset of
+AD/dementia and subsequent time to death. We fit models specifying onset of AD/dementia
+as a binary time-varying covariate, adjusting for the same time-invariant baseline covariates
+as in the above analysis (including a constant effect for APOE-ϵ4).
+For the piecewise linear effect, we set break points at 1, 2, 3, 5, and 10 years after time of
+AD onset, and for the spline effect we set 2 internal knots at observed quantiles of time from
+13
+
+AD onset to death on the log scale. Other settings and the sampling setup were as above,
+though for computation of the acceleration surface described below, we thinned the samples
+by a factor of 10 to facilitate computation. Table A.1 in Appendix A of the Supplementary
+Materials reports estimated model parameters varying baseline survival distribution and
+effect specification, along with frequentist results from an extended Cox proportional hazards
+model with AD/dementia onset as a time-varying covariate.
+As before, the coefficients
+estimated for all baseline covariates are stable across specifications for the flexible effect.
+The lefthand panels of Figure 3 show estimated regression standardized survivor curves
+comparing those without AD/dementia onset, and and those with onset at age 70 and
+85, respectively, under the TBP prior baseline specification. In each case, the curves are
+identical up until the time of onset, and then once AD/dementia onset occurs mortality
+increases substantially. The plots indicate similarity between models fit with piecewise and
+spline effects of AD/dementia onset relative to a constant effect, though the flexible models
+indicate a small delay in the mortality increase from the time of AD/dementia onset. The
+corresponding acceleration factors are given on the righthand panels of Figure 3, illustrating
+the trajectory derived in (5), where no association exists before the quantile of AD/dementia
+onset, followed by an increasingly pronounced association after AD/dementia onset.
+Selecting and plotting acceleration factors for a few AD/dementia onset times of interest
+may be sufficient in some settings, but fully communicating the results requires visualizing
+the quantile-varying effect across the entire range of the time-varying covariate. Figure 4
+reports this acceleration factor surface as a contour plot, with the time of AD/dementia onset
+on the y-axis, the survival quantile on the x-axis, and the color representing the magnitude
+of the acceleration factor. The two acceleration factor plots in Figure 3 correspond with
+cross-sections of this surface, by drawing horizontal lines at times 5 and 20 on the y-axis.
+More generally, looking horizontally across this plot shows the quantile varying acceleration
+factor corresponding with different times of AD/dementia onset. However, this plot can
+also be read vertically, to show how the acceleration factor for a particular quantile changes
+depending on the timing of the time-varying covariate. For example, drawing a vertical line
+from 0.5 on the x-axis shows the acceleration factor for median survival, varying across times
+of AD/dementia onset. Therefore, this single plot allows us to read off complex regression
+effects both as a function of the survival quantile, as well as of the timing of the time-varying
+covariate.
+6
+Discussion
+The AFT model’s specification of multiplicative covariate effects on the quantile scale pro-
+vides an interpretable and attractive alternative to the standard proportional hazards model.
+Our proposed extensions to the AFT model enabling quantile-varying acceleration factors,
+and admitting binary time-varying covariates represent important additions to the standard
+14
+
+toolbox for survival analysis. Just as the Cox proportional hazards model benefits from
+straightforward incorporation of time-varying hazard ratios, the ability to add flexibility to
+the AFT model regression effects expands the scope of scientific inquiry. Motivated by the
+study of AD in older adults, we found that the association of the APOE-ϵ4 gene with AD
+onset varied substantially across quantiles, with earlier-onset cases accelerated the most and
+later-onset cases the least.
+Moreover, the ability to model, summarize, and communicate the effects of binary time-
+varying covariates creates new opportunities to capture nuanced associations between lon-
+gitudinal health trajectories. Our proposed visualization of these effects as a surface across
+both the covariate timescale and the survival quantiles is particularly valuable, as previous
+work to incorporate time-varying covariates into AFT models has not focused on commu-
+nication of effects of time-varying components on the quantile scale (Hanson et al., 2009;
+Zhou and Hanson, 2018). In our application, this approach illustrated that the associa-
+tion between AD/dementia onset and subsequent mortality varies substantially both across
+survival quantiles, and depending on the time of AD/dementia onset.
+Estimation within the Bayesian paradigm also contributes important benefits for our
+proposed methodology. In particular, the Bayesian paradigm enables flexible estimation of
+the baseline distribution using the TBP prior, and allows for seamless uncertainty quan-
+tification even after regression standardization (Keil et al., 2018). To our knowledge, ours
+is the first implementation of the TBP prior in Stan, and software in R is available at
+https://github.com/harrisonreeder/aftquantile.
+Finally, this work complements the related literature on censored quantile regression
+(Portnoy, 2003; Reich and Smith, 2013). Censored quantile regression specifies an additive
+model for the effects of covariates on the quantile scale, while our model specifies multiplica-
+tive effects on the quantile scale. The biological plausibility or clinical relevance of additive
+versus multiplicative changes to the survival quantiles depends on the application, so our
+proposed methodology yields a valuable alternative to available quantile-based methods.
+Funding
+This project was supported by the Eunice Kennedy Shriver National Institute of Child
+Health and Human Development [grant number F31HD102159 to HTR]. The National In-
+stitutes on Aging supported the Religious Orders Study [grant numbers P30AG010161 and
+R01AG015819] and the Rush Memory and Aging Project [grant number R01AG017917].
+15
+
+Acknowledgments
+We thank the study participants and staff of the Rush Alzheimer’s Disease Center. ROSMAP
+resources can be requested at https://www.radc.rush.edu.
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+17
+
+0
+2
+4
+6
+8
+10
+0.00
+0.25
+0.50
+0.75
+1.00
+Time (t)
+Survivor Function
+V(t)
+t
+t × exp(−0.5X)
+t × exp((−0.5 + 0.6I(t > 2.5))X)
+(t1−0.5X − 1) (1 − 0.5X)
+1.00
+0.75
+0.50
+0.25
+0.00
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Survival Quantile (p)
+Acceleration Factor
+Figure 1: Sample survivor curves (left panel) and corresponding, possibly quantile-varying,
+acceleration factors (right panel). Baseline survivor function shown is S0(t) = exp(−0.3t).
+Table 1: Baseline covariates by observed AD/dementia and death outcome status.
+All
+covariates are binary, with 1 indicating the presence of the status and 0 indicating absence.
+Censored prior
+AD/dementia
+Death
+AD/dementia
+to AD/dementia
+and censored
+without
+diagnosis
+Total (%)
+or death (%)
+prior to death (%)
+AD/dementia (%)
+and death (%)
+Total
+2335 (100%)
+750 (100%)
+123 (100%)
+891 (100%)
+571 (100%)
+White Race/Ethnicity
+2178 (93.3%)
+687 (91.6%)
+100 (81.3%)
+849 (95.3%)
+542 (94.9%)
+Male Sex
+648 (27.8%)
+147 (19.6%)
+24 (19.5%)
+316 (35.5%)
+161 (28.2%)
+Married at Study Entry
+462 (19.8%)
+216 (28.8%)
+29 (23.6%)
+145 (16.3%)
+72 (12.6%)
+15+ Years of Education
+1621 (69.4%)
+532 (70.9%)
+80 (65%)
+609 (68.4%)
+400 (70.1%)
+APOE-ϵ4 Genetic Variant
+575 (24.6%)
+156 (20.8%)
+53 (43.1%)
+173 (19.4%)
+193 (33.8%)
+18
+
+Table 2: Estimated expected log predictive density (ELPD), multiplied by -2 to replicate
+scale of information criteria. Smaller values indicate better model fit.
+AFT Model
+log-Normal
+Weibull
+TBP (Weibull Centered)
+AD/Dementia Onset (Death as a Censoring Mechanism)
+Constant
+5862.0
+5806.3
+5804.4
+Piecewise Linear
+5841.5
+5788.4
+5786.5
+Restricted Cubic Spline
+5814.9
+5780.9
+5781.4
+Death (AD/Dementia as a Time-Varying Covariate)
+Constant
+9997.2
+9666.7
+9628.2
+Piecewise Linear
+9919.8
+9636.7
+9600.1
+Restricted Cubic Spline
+9884.5
+9600.9
+9564.9
+19
+
+Table 3: Regression estimates for time to onset of AD or dementia in the absence of death.
+AFT results are posterior medians and 95% credible intervals for regression parameters. Cox
+model results are log-hazard ratio estimates and 95% confidence intervals.
+AFT Model
+Cox PH
+log-Normal
+Weibull
+TBP (Weibull Centered)
+White Race/Ethnicity, β1
+Constant
+-0.28 (-0.57, 0.01)
+0.18 (0.03, 0.31)
+0.08 (-0.03, 0.19)
+0.08 (-0.03, 0.19)
+Piecewise Linear
+0.18 (0.05, 0.3)
+0.08 (-0.02, 0.17)
+0.07 (-0.03, 0.17)
+Restricted Cubic Spline
+0.15 (0.03, 0.28)
+0.07 (-0.02, 0.16)
+0.07 (-0.03, 0.17)
+Male Sex, β2
+Constant
+0.06 (-0.11, 0.23)
+-0.04 (-0.13, 0.04)
+-0.02 (-0.08, 0.05)
+-0.02 (-0.08, 0.05)
+Piecewise Linear
+-0.05 (-0.12, 0.03)
+-0.02 (-0.08, 0.04)
+-0.02 (-0.07, 0.04)
+Restricted Cubic Spline
+-0.04 (-0.11, 0.03)
+-0.02 (-0.07, 0.03)
+-0.02 (-0.07, 0.04)
+Married at Study Entry, β3
+Constant
+-0.26 (-0.48, -0.04)
+0.13 (0.03, 0.23)
+0.1 (0.02, 0.19)
+0.1 (0.03, 0.19)
+Piecewise Linear
+0.13 (0.04, 0.22)
+0.09 (0.02, 0.16)
+0.08 (0.02, 0.16)
+Restricted Cubic Spline
+0.13 (0.04, 0.22)
+0.09 (0.02, 0.16)
+0.08 (0.02, 0.16)
+≥15 Years of Education, β4
+Constant
+-0.1 (-0.26, 0.07)
+0.07 (-0.01, 0.16)
+0.04 (-0.02, 0.1)
+0.03 (-0.03, 0.09)
+Piecewise Linear
+0.07 (0, 0.15)
+0.03 (-0.02, 0.09)
+0.03 (-0.02, 0.08)
+Restricted Cubic Spline
+0.06 (-0.01, 0.14)
+0.03 (-0.02, 0.09)
+0.03 (-0.02, 0.08)
+APOE-ϵ4 Genetic Variant, β5
+Constant
+0.76 (0.61, 0.92)
+-0.42 (-0.51, -0.34)
+-0.28 (-0.35, -0.22)
+-0.28 (-0.35, -0.21)
+Piecewise Linear
+-0.79 (-0.95, -0.62)
+-0.75 (-0.92, -0.55)
+-0.76 (-0.93, -0.52)
+Restricted Cubic Spline
+-2.54 (-2.98, -1.95)
+-2.38 (-3.08, -1.23)
+-2.34 (-3.09, -0.92)
+APOE-ϵ4 Genetic Variant, α1
+Constant
+Piecewise Linear
+0.86 (0.49, 1.23)
+0.78 (0.35, 1.20)
+0.78 (0.28, 1.23)
+Restricted Cubic Spline
+1.51 (1.12, 1.85)
+1.51 (0.77, 2.01)
+1.5 (0.61, 2.03)
+APOE-ϵ4 Genetic Variant, α2
+Constant
+Piecewise Linear
+0.52 (0.23, 0.80)
+0.74 (0.39, 1.06)
+0.79 (0.41, 1.14)
+Restricted Cubic Spline
+3.83 (2.73, 4.47)
+3.63 (1.45, 4.76)
+3.52 (0.87, 4.78)
+APOE-ϵ4 Genetic Variant, α3
+Constant
+Piecewise Linear
+0.46 (0.11, 0.81)
+0.97 (0.59, 1.34)
+0.99 (0.58, 1.39)
+Restricted Cubic Spline
+1.07 (0.71, 1.4)
+1.34 (0.77, 1.77)
+1.32 (0.65, 1.79)
+APOE-ϵ4 Genetic Variant, α4
+Constant
+Piecewise Linear
+-0.38 (-1.06, 0.45)
+0.41 (-0.23, 1.20)
+0.37 (-0.35, 1.20)
+Restricted Cubic Spline
+20
+
+0
+10
+20
+30
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time to AD/Dementia without Death, Years from Age 65
+Survivor Function
+No APOE-e4
+PH, Constant
+AFT, Constant
+AFT, Piecewise Linear
+AFT, Spline
+APOE-e4
+PH, Constant
+AFT, Constant
+AFT, Piecewise Linear
+AFT, Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+Figure 2: Under a Weibull-centered TBP baseline specification: (left panel) regression stan-
+dardized survivor function estimates for onset of AD or dementia without death, averaged
+over other covariates.
+Regression standardized estimate from Cox proportional hazards
+model shown for comparison; (right panel) regression standardized quantile-varying accel-
+eration factor estimates for onset of AD or dementia without death, averaged over other
+covariates. 95% credible intervals represented with dashed lines. Grey shaded region repre-
+sents area of parametric extrapolation beyond quantiles observed in both groups.
+21
+
+0
+10
+20
+30
+40
+0.0
+0.4
+0.8
+Survivor Function
+AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+No AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+0
+10
+20
+30
+40
+0.0
+0.4
+0.8
+Time to Death, Years from Age 65
+Survivor Function
+AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+No AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.6
+1.0
+Survival Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.6
+1.0
+Survival Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+Figure 3: Under a Weibull-centered TBP baseline specification: (left panel) regression stan-
+dardized survivor function estimates for mortality following onset of AD or dementia, aver-
+aged over other covariates; (Right panel) regression standardized survivor function estimates
+for mortality following onset of AD or dementia, averaged over other covariates. 95% credible
+intervals represented with dashed lines. Grey shaded region represents area of parametric
+extrapolation beyond quantiles observed in both groups.
+22
+
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+(0.45, 0.50]
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(1.20, 1.25]
+(1.15, 1.20]
+(1.10, 1.15]
+(1.05, 1.10]
+(1.00, 1.05]
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+Figure 4: Under a Weibull-centered TBP baseline specification, contour plots of regression
+standardized acceleration factor surface estimates for death following onset of AD/dementia,
+standardized to other covariates. Time of AD/dementia onset is shown on y-axis, and sub-
+sequent survival quantile is shown on x-axis.
+Color indicates acceleration factor at the
+given survival quantile, comparing those with AD/dementia onset at the specified time and
+those without AD/dementia. Horizontal cross-sections illustrate quantile-varying accelera-
+tion factor for AD/dementia onset at a particular time, while vertical cross-sections illustrate
+acceleration factor at a particular quantile across times of AD/dementia onset. (Left panel)
+constant effect specification; (center panel) piecewise linear effect specification; (right panel)
+spline effect specification.
+23
+
+Appendix Introduction
+In this appendix we present additional details and results beyond what could be presented
+in the main manuscript. To distinguish the two documents, alpha-numeric labels are used in
+this document while numeric labels are used in the main paper. Section A provides additional
+results from the data application. Section B provides derivation of the form of V −1 when V
+is specified as a piecewise linear function of time. Section C provides derivation of the form of
+the acceleration factor associated with a binary time-varying covariate. Section D provides
+additional detail on the transformed Bernstein polynomial (TBP) prior specification.
+24
+
+A
+Additional Data Application Results
+A.1
+AD/Dementia Onset
+In this section we report additional regression-standardized survival curves and acceleration
+factors for the onset of AD or dementia by APOE-ϵ4 genetic variant status, for alternative
+specifications for the baseline distribution. We note that the most substantial difference be-
+tween specifications occurs in the lowest quantiles, which represent parametric extrapolation
+beyond the observed data quantiles.
+0
+10
+20
+30
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time to AD/Dementia without Death, Years from Age 65
+Survivor Function
+No APOE-e4
+PH, Constant
+AFT, Constant
+AFT, Piecewise Linear
+AFT, Spline
+APOE-e4
+PH, Constant
+AFT, Constant
+AFT, Piecewise Linear
+AFT, Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+Figure A.1: Under a Weibull baseline specification: (left panel) regression standardized
+survivor function estimates for onset of AD or dementia without death, averaged over other
+covariates. Regression standardized estimate from Cox proportional hazards model shown
+for comparison; (right panel) regression standardized quantile-varying acceleration factor
+estimates for onset of AD or dementia without death, averaged over other covariates. 95%
+credible intervals represented with dashed lines.
+Grey shaded region represents area of
+parametric extrapolation beyond quantiles observed in both groups.
+25
+
+0
+10
+20
+30
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time to AD/Dementia without Death, Years from Age 65
+Survivor Function
+No APOE-e4
+PH, Constant
+AFT, Constant
+AFT, Piecewise Linear
+AFT, Spline
+APOE-e4
+PH, Constant
+AFT, Constant
+AFT, Piecewise Linear
+AFT, Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+Figure A.2: Under a log-Normal baseline specification: (left panel) regression standardized
+survivor function estimates for onset of AD or dementia without death, averaged over other
+covariates. Regression standardized estimate from Cox proportional hazards model shown
+for comparison; (right panel) regression standardized quantile-varying acceleration factor
+estimates for onset of AD or dementia without death, averaged over other covariates. 95%
+credible intervals represented with dashed lines.
+Grey shaded region represents area of
+parametric extrapolation beyond quantiles observed in both groups.
+26
+
+A.2
+Mortality following AD/Dementia Onset
+Below we report regression parameter estimates, and additional regression-standardized sur-
+vival curves and acceleration factors for mortality by AD/dementia status, across alternative
+specifications for the baseline distribution.
+0
+10
+20
+30
+40
+0.0
+0.4
+0.8
+Survivor Function
+AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+No AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+0
+10
+20
+30
+40
+0.0
+0.4
+0.8
+Time to Death, Years from Age 65
+Survivor Function
+AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+No AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.6
+1.0
+Survival Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.6
+1.0
+Survival Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+Figure A.3: Under a Weibull baseline specification: (left panel) regression standardized sur-
+vivor function estimates for mortality following onset of AD or dementia, averaged over other
+covariates; (Right panel) regression standardized survivor function estimates for mortality
+following onset of AD or dementia, averaged over other covariates. 95% credible intervals
+represented with dashed lines. Grey shaded region represents area of parametric extrapola-
+tion beyond quantiles observed in both groups.
+27
+
+0
+10
+20
+30
+40
+0.0
+0.4
+0.8
+Survivor Function
+AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+No AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+0
+10
+20
+30
+40
+0.0
+0.4
+0.8
+Time to Death, Years from Age 65
+Survivor Function
+AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+No AD/Dementia Onset
+Constant
+Piecewise Linear
+Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.6
+1.0
+Survival Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.6
+1.0
+Survival Quantile (p)
+Acceleration Factor
+Constant
+Piecewise Linear
+Spline
+Figure A.4: Under a log-Normal baseline specification: (left panel) regression standard-
+ized survivor function estimates for mortality following onset of AD or dementia, averaged
+over other covariates; (Right panel) regression standardized survivor function estimates for
+mortality following onset of AD or dementia, averaged over other covariates. 95% credible
+intervals represented with dashed lines. Grey shaded region represents area of parametric
+extrapolation beyond quantiles observed in both groups.
+28
+
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+(0.45, 0.50]
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+(0.45, 0.50]
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(1.05, 1.10]
+(1.00, 1.05]
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+(0.45, 0.50]
+Figure A.5: Under a Weibull baseline specification, contour plots of regression standardized
+acceleration factor surface estimates for death following onset of AD/dementia, standard-
+ized to other covariates. Time of AD/dementia onset is shown on y-axis, and subsequent
+survival quantile is shown on x-axis. Color indicates acceleration factor at the given survival
+quantile, comparing those with AD/dementia onset at the specified time and those without
+AD/dementia. Horizontal cross-sections illustrate quantile-varying acceleration factor for
+AD/dementia onset at a particular time, while vertical cross-sections illustrate acceleration
+factor at a particular quantile across times of AD/dementia onset. (Left panel) constant
+effect specification; (center panel) piecewise linear effect specification; (right panel) spline
+effect specification.
+29
+
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+(0.45, 0.50]
+(0.40, 0.45]
+(0.35, 0.40]
+0
+10
+20
+30
+40
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+(0.45, 0.50]
+(0.40, 0.45]
+(0.35, 0.40]
+(0.30, 0.35]
+(0.25, 0.30]
+(0.20, 0.25]
+0
+10
+20
+30
+0.00
+0.25
+0.50
+0.75
+1.00
+Survival Quantile (p)
+Years since 65 at AD Onset
+AF
+(1.05, 1.10]
+(1.00, 1.05]
+(0.95, 1.00]
+(0.90, 0.95]
+(0.85, 0.90]
+(0.80, 0.85]
+(0.75, 0.80]
+(0.70, 0.75]
+(0.65, 0.70]
+(0.60, 0.65]
+(0.55, 0.60]
+(0.50, 0.55]
+(0.45, 0.50]
+(0.40, 0.45]
+(0.35, 0.40]
+(0.30, 0.35]
+(0.25, 0.30]
+(0.20, 0.25]
+Figure A.6: Under a log-Normal baseline specification, contour plots of regression standard-
+ized acceleration factor surface estimates for death following onset of AD/dementia, stan-
+dardized to other covariates. Time of AD/dementia onset is shown on y-axis, and subsequent
+survival quantile is shown on x-axis. Color indicates acceleration factor at the given survival
+quantile, comparing those with AD/dementia onset at the specified time and those without
+AD/dementia. Horizontal cross-sections illustrate quantile-varying acceleration factor for
+AD/dementia onset at a particular time, while vertical cross-sections illustrate acceleration
+factor at a particular quantile across times of AD/dementia onset. (Left panel) constant
+effect specification; (center panel) piecewise linear effect specification; (right panel) spline
+effect specification.
+30
+
+Table A.1: Regression estimates for time to death. AFT results are posterior medians and
+95% credible intervals for regression parameters. Cox model results are log-hazard ratio
+estimates and 95% confidence intervals.
+AFT Model
+Cox PH
+log-Normal
+Weibull
+TBP (Weibull Centered)
+White Race/Ethnicity, β1
+Constant
+0.17 (-0.07, 0.41)
+-0.07 (-0.17, 0.03)
+-0.08 (-0.16, 0)
+-0.07 (-0.14, 0)
+Piecewise Linear
+-0.1 (-0.21, 0.01)
+-0.08 (-0.16, 0)
+-0.07 (-0.15, 0.01)
+Restricted Cubic Spline
+-0.1 (-0.21, 0)
+-0.08 (-0.16, 0)
+-0.07 (-0.15, 0)
+Male Sex, β2
+Constant
+0.52 (0.41, 0.64)
+-0.25 (-0.31, -0.2)
+-0.16 (-0.2, -0.12)
+-0.14 (-0.17, -0.11)
+Piecewise Linear
+-0.27 (-0.33, -0.21)
+-0.16 (-0.2, -0.13)
+-0.14 (-0.18, -0.11)
+Restricted Cubic Spline
+-0.27 (-0.33, -0.21)
+-0.16 (-0.2, -0.13)
+-0.14 (-0.18, -0.11)
+Married at Study Entry, β3
+Constant
+-0.16 (-0.3, -0.01)
+0.12 (0.05, 0.19)
+0.05 (0.01, 0.1)
+0.04 (0, 0.08)
+Piecewise Linear
+0.12 (0.05, 0.19)
+0.05 (0.01, 0.1)
+0.04 (0, 0.08)
+Restricted Cubic Spline
+0.12 (0.05, 0.19)
+0.05 (0.01, 0.1)
+0.04 (0, 0.08)
+≥15 Years of Education, β4
+Constant
+-0.1 (-0.21, 0.01)
+0.07 (0.02, 0.13)
+0.03 (-0.01, 0.06)
+0.02 (-0.01, 0.05)
+Piecewise Linear
+0.09 (0.03, 0.15)
+0.03 (-0.01, 0.07)
+0.02 (-0.01, 0.06)
+Restricted Cubic Spline
+0.09 (0.03, 0.15)
+0.03 (0, 0.07)
+0.02 (-0.01, 0.06)
+APOE-ϵ4 Genetic Variant, β5
+Constant
+0.01 (-0.11, 0.13)
+0.06 (0, 0.11)
+0.01 (-0.03, 0.05)
+0 (-0.03, 0.04)
+Piecewise Linear
+0.06 (0, 0.12)
+0.01 (-0.03, 0.05)
+0 (-0.03, 0.04)
+Restricted Cubic Spline
+0.06 (0, 0.12)
+0.01 (-0.03, 0.05)
+0 (-0.03, 0.04)
+AD/Dementia Onset, β6
+Constant
+1.14 (1.02, 1.26)
+-1.06 (-1.16, -0.97)
+-0.73 (-0.81, -0.65)
+-0.68 (-0.76, -0.61)
+Piecewise Linear
+-0.14 (-0.42, 0.17)
+-0.02 (-0.3, 0.29)
+0.01 (-0.27, 0.32)
+Restricted Cubic Spline
+1.59 (0.90, 2.35)
+1.63 (0.95, 2.38)
+1.68 (1, 2.42)
+AD/Dementia Onset, α1
+Constant
+Piecewise Linear
+-0.95 (-1.29, -0.63)
+-0.86 (-1.21, -0.53)
+-0.84 (-1.18, -0.53)
+Restricted Cubic Spline
+-1.77 (-2.25, -1.33)
+-1.60 (-2.08, -1.17)
+-1.57 (-2.05, -1.14)
+AD/Dementia Onset, α2
+Constant
+Piecewise Linear
+-1.15 (-1.49, -0.82)
+-0.90 (-1.25, -0.57)
+-0.86 (-1.21, -0.54)
+Restricted Cubic Spline
+-5.07 (-6.56, -3.73)
+-4.40 (-5.86, -3.10)
+-4.43 (-5.87, -3.11)
+AD/Dementia Onset, α3
+Constant
+Piecewise Linear
+-1.15 (-1.49, -0.82)
+-0.65 (-0.99, -0.33)
+-0.62 (-0.96, -0.31)
+Restricted Cubic Spline
+-1.78 (-2.18, -1.41)
+-1.08 (-1.45, -0.75)
+-1.08 (-1.44, -0.74)
+AD/Dementia Onset, α4
+Constant
+Piecewise Linear
+-1.68 (-2.11, -1.26)
+-0.71 (-1.10, -0.32)
+-0.74 (-1.15, -0.34)
+Restricted Cubic Spline
+31
+
+B
+Derivation of V −1(t | X) under Piecewise Linearity
+Under piecewise linear specification of V (t | X), define J + 2 knots 0 = τ0 < τ1 < · · · < τJ <
+τJ+1 = ∞, with piecewise linear basis functions defined Bj(t | τ) = t−1(min{t, τj+1} − τj)+
+where (z)+ = min{0, z}. Assuming a flexible effect for X1, the resulting specification becomes
+V (t | X) = t × exp (−X
+Tβ)
+� J
+�
+j=1
+exp (−X1αj) Bj(t | τ)
+�
+.
+The inverse function V −1 can be derived by inspection, noting that the inverse of an increas-
+ing piecewise linear function is also an increasing piecewise linear function, with changepoints
+shifted according to the values of X, β, and α. Specifically, define τ ∗ such that τ ∗
+0 = τ0 = 0,
+τ ∗
+1 = τ1 × exp(−XTβ), and for j > 1,
+τ ∗
+j = τ ∗
+1 +
+j
+�
+l=2
+exp(−X
+Tβ − X1αl−1)(τl − τl−1).
+The lines on each interval of V −1 have the inverse slope of the line in the corresponding
+interval of V , so the final inverse function is succinctly written
+V −1(t | X) = t × exp (X
+Tβ)
+� J
+�
+j=1
+exp (X1αj) Bj(t | τ ∗)
+�
+.
+C
+Derivation of Acceleration Factor for a Binary Time-
+Varying Covariate
+Let X1(t) be a binary-valued step function, such as an indicator for whether a non-terminal
+event has occurred by time t. Formally, define X1(t) = I(t > tX), where tX is the time at
+which X1 changes. Consider a single additional covariate time-invariant covariate X2.
+Notating t∗
+X = tX exp(−X2β2), the inverse function for V as defined in (5) is derived
+following Appendix B as
+V −1(t | X(t)) = exp (X2β2) [min{t, t∗
+X} + (t − t∗
+X)+ exp (β1)] .
+The resulting acceleration factor at quantile p between a person with X2 = x2 who experi-
+ences the non-terminal event at time tX, and a person with X2 = x′
+2 who experiences the
+non-terminal event at time t′
+X, is
+ξ(p | tX, t′
+X, x2, x′
+2, S0) = e(x2−x′
+2)β2 min{S−1
+0 (p), tXe−x2β2} + eβ1(S−1
+0 (p) − tXe−x2β2)+
+min{S−1
+0 (p), t′
+Xe−x′
+2β2} + eβ1(S−1
+0 (p) − t′
+Xe−x′
+2β2)+
+.
+Finally, note that when a general flexible effect for X1(t) is specified, in general no closed
+form exists, but acceleration factors can still be computed numerically.
+32
+
+D
+Transformed Bernstein Polynomial Prior
+To illustrate the flexibility of the transformed Bernstein polynomial prior, Figure D.1 shows
+the basis functions when K = 5,
+G(p | k, K − k + 1) =
+Γ(K + 1)
+Γ(k)Γ(K − k + 1)pk−1(1 − p)K−k.
+Moreover, Figure D.2 shows a sample of different shapes that the resulting baseline survivor
+function S0(t | φ, w) = �K
+k=1 wkG(S∗
+0(t | φ) | k, K − k + 1) can take, for selected weight
+vectors w and setting S∗
+0(t) = exp(−t).
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.4
+0.8
+p
+G(p)
+Figure D.1: Basis functions G for j = 1, . . . , 5.
+33
+
+0
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t
+S0(t)
+1
+2
+3
+4
+1
+2
+3
+4
+w1
+0.01
+0.64
+0.07
+0.41
+w2
+0.03
+0.23
+0.18
+0.02
+w3
+0.09
+0.09
+0.50
+0.01
+w4
+0.23
+0.03
+0.18
+0.15
+w5
+0.64
+0.01
+0.07
+0.41
+Figure D.2: Sample survivor functions corresponding to varying transformed bernstein poly-
+nomial prior weight vectors. Bold black line shows centering distribution S∗
+0(t) = exp(−t).
+34
+

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+The average degree of edge chromatic critical graphs with
+maximum degree seven
+Yan Cao
+Scdool of Mathematical Sciences, Dalian University of Technology
+Dalian, Liaoning, 116024, China
+Email: [email protected]
+Rong Luo∗
+Department of Mathematics, West Virginia University
+Morgantown, WV 26505
+Email: [email protected]
+Zhengke Miao†
+School of Mathematics and Statistics, Jiangsu Normal University
+Xuzhou, Jiangsu, 221116, China
+Email: [email protected]
+Yue Zhao
+Department of Mathematics, University of Central Florida
+Orlando, FL 32816-1364
+Email: [email protected]
+Abstract
+In this paper, by developing several new adjacency lemmas about a path on 4 or 5 ver-
+tices, we show that the average degree of 7-critical graphs is at least 6. It implies Vizing’s
+planar graph conjecture for planar graphs with maximum degree 7 and its extension to
+graphs embeddable in a surface with nonnegative Euler characteristic due to Sanders and
+Zhao (J. Combin. Theory Ser. B 83 (2001) 201-212 and J. Combin. Theory Ser. B 87
+(2003) 254-263) and Zhang (Graphs and Combinatorics 16 (2000) 467-495).
+Keywords:. Edge coloring, critical graphs, Euler’s formula, planar graphs
+1
+Introduction
+An edge coloring of a graph is a function assigning values (colors) to the edges of the graph in
+such a way that any two adjacent edges receive different colors. A graph is edge k-colorable if
+there is an edge coloring of the graph with colors from C = {1, . . . , k}. A finite simple graph
+∗Partially supported by a grant from Simons Foundation (No. 839830)
+†Partially supported by NSFC under grant numbers 12031018 and 11971205.
+1
+arXiv:2301.02140v1  [math.CO]  5 Jan 2023
+
+G of maximum degree ∆ is class one if it is edge ∆-colorable. Otherwise, G is said to be
+class two, in which case Vizing’s Theorem [20] guarantees that it is edge (∆ + 1)-colorable.
+G is said to be edge chromatic critical (or critical for short) if it is connected, class two and
+χ′(G − e) < χ′(G) for every edge e ∈ G. A critical graph G of maximum degree ∆ is called a
+∆-critical graph. Vizing proposed the following conjecture in 1968 [21] on the average degree
+of ∆-critical graphs.
+Conjecture 1.1 Let G be a ∆-critical graph. Then d(G) ≥ ∆ − 1 +
+3
+|V (G)|, where d(G) is the
+average degree of G.
+There are direct consequences of a progress towards solving this conjecture. For example,
+if there is a better bound for the size of ∆-critical graphs, then one can obtain better bounds
+for ∆(S), where S is a surface and ∆(S) = max{∆(G)|G is a class two connected graph that
+can be embedded in S}. It is well known that if Vizing’s conjecture is true for ∆ = 7, then
+∆(S) ≤ 6 where S is a surface of Euler characteristic at least 1, which was proved in [17] by
+other means in 2003. If this average degree conjecture is true, for a ∆-critical graph G, by
+applying the inequality α ≤ n− m
+∆, where n = |V (G)|, m = |E(G)|, and α is the independence
+number of G, one can easily obtain α ≤ n
+2 as ∆ → ∞. This provides a strong evidence for the
+independence number conjecture proposed by Vizing in 1968 [21], which claims that if G is a
+critical graph, then α ≤ n
+2 .
+Conjecture 1.1 was verified for ∆ = 3 by Jakobsen [12], for ∆ = 4 by Fiorini and Wilson
+[10], for ∆ = 5 by Kayathri [13], and for ∆ = 6 by Luo, Miao and Zhao [14]. As for the lower
+bound of d(G), Woodall [22] proved that if G is a ∆-critical graph, then d(G) ≥ 2(∆+1)
+3
+. Cao
+and Chen [5] further improved to 3∆
+4 − 8 and they [5, 6] also showed that Conjecture 1.1 is
+asymptotically true.
+In this paper, we will prove that if G is a 7-critical graph, then d(G) ≥ 6. This result implies
+Vizing’s planar graph conjecture for ∆ = 7 claiming that every planar graph with maximum
+degree at least 7 is class one, which was verified independently by Sanders and Zhao [17]
+and Zhang [23] and its extension to graphs embeddable in a surface with nonnegative Euler
+characteristic due to Sanders and Zhao in [17] and [18].
+Before proceeding, we introduce some notations. Throughout this paper, let G = (V, E)
+be a simple graph with n vertices, m edges, and maximum degree ∆(G) (or ∆). A k-vertex,
+k+-vertex, or k−-vertex is a vertex of degree k, at least k, or at most k, respectively. We
+use d(x), dk(x), dk+(x), dk−(x) to denote the degree of a vertex x, the number of k-vertices
+adjacent to x, the number of k+-vertices adjacent to x, and the number of k−-vertices adjacent
+to x, respectively. For a vertex v ∈ V , let N(x) = {v|xv ∈ E} be the neighborhood of v in
+G. A k-neighbor of a vertex v is a neighbor of v that is a k-vertex in G, a k+-neighbor or
+k−-neighbor of a vertex v is a neighbor of v that is a k+-vertex or k−-vertex in G. For two
+disjoint vertex sets U and U ′, denote by [U, U′] the set of edges with one end in U and the
+other in U ′. For a vertex set A of V (G), denote by N(A) = ∪x∈AN(x).
+2
+
+2
+Lemmas
+In this section, we present some old lemmas and develop some new lemmas needed in the
+proofs of our main result.
+2.1
+Old lemmas
+Lemma 2.1 (Vizing’s Adjacency Lemma [20]) Let G be a ∆-critical graph. Then d(u) +
+d(v) ≥ ∆ + 2 for any two adjacent vertices u and v, and d∆(x) ≥ max{2, ∆ − k + 1} if x has
+a k-neighbor.
+Lemma 2.2 (Luo, Miao, and Zhao [14]) Let G be a ∆-critical graph with ∆ ≥ 5 and x be a
+3-vertex. Then x has at least two ∆-neighbors which are not adjacent to any (∆−2)−-vertices
+except x.
+Lemma 2.3 (Luo, Miao, and Zhao [16]) Let G be a ∆-critical graph with ∆ ≥ 6 and x be a
+3-vertex. Then x has a ∆-neighbor which is adjacent to at least ∆ − 4 − ⌊ ∆−1
+3 ⌋ vertices z with
+d(z) = ∆ and d(∆−3)−(z) = 0.
+Lemma 2.4 (Sanders and Zhao [17] and Zhang [23]) Let G be ∆-critical graph and xyrs be a
+path with d(x) + d(y) = ∆ + 2. Then d(r) = ∆ and d(s) ≥ ∆ − 1. Moreover if d(x), d(y) < ∆,
+then d(s) = ∆.
+Lemma 2.5 (Luo, Miao, and Zhao [14]) Let G be a ∆-critical graph with ∆ ≥ 6 and x be a
+4-vertex.
+(1) If x is adjacent to a (∆ − 2)-vertex, say y, then N(N(x)) \ {x, y} ⊆ V∆;
+(2) If x is not adjacent to any (∆ − 2)-vertex and if one of the neighbors y of x is adjacent to
+d(y) − (∆ − 3) vertices of degree at most ∆ − 2, then each of the other three neighbors of x is
+adjacent to only one (∆ − 2)−-vertex, which is x;
+(3) If x is adjacent to two (∆ − 1)-vertices, then each of the neighbors of x is adjacent to
+exactly one (∆ − 2)−-vertex, which is x.
+The following lemma is a special case of Lemma 2.4 in [17] due to Sanders and Zhao.
+Lemma 2.6 Let G be a 7-critical graph and xyz be a path in G. If 3 ≤ d(x) ≤ 4, d(y) = 7
+and d(x) + d(z) ≤ 8, then y and z have at most d(x) − 3 common neighbors.
+2.2
+New lemmas
+The following lemmas will be proved in Section 5.
+Let G be a ∆-critical graph. For each vertex v, denote
+N∆∼2(v) = {z ∈ N(v) : z has a neighbor of degree 2}
+Lemma 2.7 Let G be a ∆-critical graph with ∆ ≥ 7. Then |N∆∼2(v)| ≤ 5 for every v ∈ V (G).
+Lemma 2.8 Let G be a ∆-critical graph and xyrst be a path with d(x) + d(y) = ∆ + 2 and
+max{d(x), d(y)} < ∆. Then d(t) ≥ ∆ − 2.
+Lemma 2.9 Let G be a ∆-critical graph and xyrst be a path with d(x) = 3 and d(y) = ∆.
+Suppose that y has a neighbor z ̸∈ {x, r, s} with d(z) ≤ ∆ − 2. Then d(s) ≥ ∆ − 1; and
+d(z) + d(t) ≥ ∆ + 1 if d(t) ≤ ∆ − 4.
+3
+
+So far all adjacency lemmas are about a path on at most four vertices. Lemma 2.9 is the
+first lemma that deals with a path with five vertices.
+By Lemmas 2.4, 2.8, and 2.9, we have the following corollary.
+Corollary 2.10 Let G be a 7-critical graph and xyrst be a path with d(x) = 3. Then we have
+the following:
+(1) if d(y) = 6, then d(r) = d(s) = 7 and d(t) ≥ 5.
+(2) if d(y) = 7 and y has another 4−-neighbor other than x, then d(s) ≥ 6 and d(t) ≥ 4.
+(3) if d(y) = 7 and y has a 5-neighbor, then either d(s) = 6 and d(t) ≥ 4 or d(s) = 7 and
+d(t) ≥ 3.
+Lemma 2.11 Let G be a ∆-critical graph and xy be an edge with d(x) + d(y) = ∆ + 3 and
+max{d(x), d(y)} < ∆. Then x has d(x)−2 neighbors of degree ∆ having no (∆−2)−-neighbors
+other than x, y.
+Lemma 2.12 Let G be a 7-critical graph and x be a 5-vertex.
+(1) if x has three 6-neighbors, then each 7-neighbor of x has exactly one 5−-neighbor.
+(2) if x has two 6-neighbors, then x has two 7-neighbors, each of which has at most two
+5−-neighbors.
+(3) if x has exactly four 7-neighbors, then x has two 7-neighbors, each of which has at most
+three 5−-neighbors.
+3
+The average degree of 7-critical graphs
+3.1
+Main result
+In this section we will prove our main result.
+Theorem 3.1 d(G) ≥ 6 for every 7-critical graph G.
+Proof. Let G be a 7-critical graph. We define the following subsets of vertices.
+A = {u|d(u) = 7 and u is adjacent to a 2-vertex},
+B = {u|d(u) = 6 and u is adjacent to a 3-vertex},
+C = {u|d(u) = 7 and u is adjacent to a 3-vertex and a 5−-vertex}.
+The following proposition is straightforward from Lemma 2.7 and Corollary 2.10.
+Proposition 3.2 Let x be a 7-vertex which is not adjacent to a 5−-vertex. Then at most one
+of the three sets N(x)∩A, N(x)∩B, and N(x)∩C is a nonempty set. Moreover |N(x)∩A| ≤ 5
+and |N(x) ∩ B| ≤ 1.
+For each vertex x, denote by M(x) = d(x) − 6 to be the initial charge of x.
+R1 Let u be a 7-vertex not adjacent to a 5−-vertex but adjacent to a vertex in A ∪ B ∪ C.
+Then u sends
+1
+|N(x)∩A|+|N(x)∩B|+|N(x)∩C| to each neighbor in A ∪ B ∪ C.
+R2 Let u be a 7-vertex adjacent to a 5−-vertex. Then u sends
+1
+d5−(u) to each neighbor with
+degree 4 or 5, 1 to each 3-neighbor, and 2 to each 2-neighbor.
+R3 Every 6-vertex sends 1 to each 3-neighbor.
+4
+
+R4 If a 5-vertex u is adjacent to a 7-vertex v ∈ C, then u sends 1
+8 to v.
+R5 If a 4-vertex is adjacent to a 5-vertex, then the 4-vertex receives 1
+2 from its 5-neighbor.
+Denote by M′(x) to be the new charge of the vertex x. We have the following estimation
+for M′(x).
+(I) Let u be a vertex with degree 2 or 3. Then M′(u) = 0.
+By (R2), each 2-vertex receives 2 from each neighbor. By Lemma 2.1, each 3-vertex is
+not adjacent to a 5−-vertex. Thus by (R2) and (R3), each 3-vertex receives 1 from each
+neighbor. Therefore M′(u) = 0 if d(u) = 2 or 3.
+(II) Let uv be an edge with d(u) + d(v) = ∆ + 2 = 9 and 3 ≤ d(u) ≤ d(v) < 7. Then
+M′(u) ≥ 0 and M′(v) ≥ 1.
+Let w ∈ N(u)∪N(v) and w ̸∈ {u, v}. If w ∈ N(u)∩N(v), then by Lemma 2.4, d(w) = 7,
+and w has only two 6−-neighbors. Thus by (R2), w sends 1
+2 to each of u and v if d(u) = 4
+and d(v) = 5 and sends 1 to u, 0 to v if d(u) = 3 and d(v) = 6.
+If w ̸∈ N(v) ∩ N(u), then by Lemma 2.4, d(w) = 7 and w has only one 6−-neighbor,
+which is either u or v. If w ∈ N(u), then by (R2), w sends 1 to u. Assume w ∈ N(v).
+If d(v) = 6, then v ∈ B, and by Proposition 3.2, w sends 1 to v. If d(v) = 5, then
+N(w) ∩ (A ∪ B ∪ C) = ∅ by Lemma 2.8 and thus w sends 1 to v by (R2). Therefore in
+any case w sends 1 to either u or v if w ̸∈ N(v) ∩ N(u).
+If d(u) = 4 and d(v) = 5, then u receives 1
+2 from v by (R2). Thus M′(u) ≥ 4−6+4× 1
+2 = 0
+and M′(v) = 5 − 6 + 1
+2|N(u) ∩ N(v)| + |N(v) \ N(u)| − 1
+2 ≥ 5 − 6 + 3
+2 + 1 − 1
+2 ≥ 1.
+If d(u) = 3 and d(v) = 6, then M′(u) = 0 by (I) and v sends 1 to u by (R3). Thus
+M′(v) = 6 − 6 + |N(v) \ N(u)| − 1 ≥ 6 − 6 + 3 − 1 > 1.
+(III) Let u be a 4-vertex with four 6+-neighbors. Then M′(u) > 0 unless u has either four
+7-neighbors or has two 6-neighbors and two 7-neighbors, in which case M′(u) ≥ 0.
+By Lemma 2.1, u is adjacent to at least two 7-vertices and each 7-neighbor of u is
+adjacent to at most three 5−-vertices.
+If u has a 7-neighbor v adjacent to three 5−-vertices, then by Lemma 2.5, u is adjacent to
+four 7-vertices and except v, each 7-neighbor is adjacent to only one 5−-vertex. Therefore
+by (R2), M′(u) ≥ 4 − 6 + 3 × 1 + 1
+3 = 4
+3.
+Now assume that each 7-neighbor is adjacent to at most two 5−-vertices. Then u receives
+at least 1
+2 from each 7-neighbor.
+If u has four 7-neighbors, then M′(u) ≥ 4 − 6 + 4 × 1
+2 = 0.
+If u has a 6-neighbor, then by Lemma 2.11, there are two 7-neighbors of u having only
+one 5−-neighbor. Thus M′(u) ≥ −2 + 2 + 1
+2(d7(u) − 2) ≥ 0 with equality when u has
+exactly two 6-neighbors and two 7-neighbors.
+5
+
+(IV) M′(u) > 0 for each 5-vertex u with five 5+-neighbors.
+By Lemma 2.1, u is adjacent to at least two 7-vertices and each 7-neighbor of u is
+adjacent to at most four 6−-vertices.
+If v is a 7-neighbor of u and v is adjacent to a 3-vertex, then v sends 1
+2 to u by (R2)
+and u sends 1
+8 to v by (R4). Therefore the total net charge u receives from v is 3
+8.
+Thus in general, u receives at least min{ 3
+8, 1
+4} from each 7-neighbor.
+If u has at least four 7-neighbors, then by Lemma 2.12(3), M′(u) ≥ −1+2× 1
+4 +2× 1
+3 > 0.
+Now assume that u is adjacent to at most three 7-vertices.
+If u is adjacent to a 5-vertex, then by Lemma 2.11, u has three 7-neighbors, each of
+which could be adjacent to at most two 5−-vertex (u and the 5-neighbor of u). Thus
+M′(u) ≥ −1 + 3 × 1
+2 = 1
+2 > 0.
+Finally, we may assume that u is adjacent to at least two 6-vertices and at most three
+7-vertices. By Lemma 2.12(1) and (2), M′(u) ≥ −1 + min{1
+4 + 2 × 1
+2, 1 + 1} > 0.
+(V) Let u be a 6-vertex adjacent to six 4+-vertices. Then by the discharging rules, M′(u) =
+M(u) = 0.
+(VI) M′(u) ≥ 0 if d(u) = 7.
+Let u be a 7-vertex. Then u ̸∈ B. By (R1) and (R2), we have M′(u) ≥ 0 if u ̸∈ A ∪ C.
+(a) Assume u ∈ A (that is u has a 2-neighbor v).
+Let w be the other neighbor of v and x ∈ N(u) \ {v, w}. Then by Lemma 2.4, d(x) = 7
+and x is not adjacent to a 5−-vertex. Since u ∈ A, by Proposition 3.2, x is adjacent to at
+most five vertices in A∪C. Thus by (R1), x sends at least 1
+5 to u. Since |N(u)\{v, w}| ≥
+5, we have M′(u) ≥ 7 − 6 − 2 + 5 × 1
+5 = 0.
+(b) Assume u ∈ C (that is u is adjacent to a 3-vertex x and another 5−-vertex z).
+By Lemma 2.1, x and z are not adjacent and u has five 7-neighbors. By Lemma 2.6, u
+and z have no common neighbor. Thus u has at least three 7-neighbors which are not
+adjacent to x or z. Let w be such a 7-neighbor of u. By Proposition 3.2, N(w)∩(A∪B) =
+∅.
+If d(z) ≤ 4, then 3 ≤ d(z) ≤ 4 by Lemma 2.1, and thus u sends at most 1 to each of x
+and z. By Corollary 2.10(2), u is the only vertex in C adjacent to w. So w sends 1 to u
+by (R1). Thus M′(u) ≥ 7 − 6 − 1 − 1 + 3 = 1.
+If d(z) = 5, then w is adjacent to at most seven vertices in C and thus sends at least 1
+7
+to u by (R1). By (R2), u sends 1 to x and 1
+2 to z and by (R4), z sends 1
+8 to u. Therefore
+M′(u) ≥ 7 − 6 − 1 − 1
+2 + 1
+8 + 3
+7 > 0. This completes the proof of (VI).
+By (I)-(VI), M′(x) ≥ 0 for each vertex x and thus 0 ≤ �
+x∈V M′(x) = �
+x∈V M(x) =
+(d(G) − 6)|V |. Therefore d(G) ≥ 6. This completes the proof of the theorem.
+6
+
+3.2
+Concluding remarks
+One may wonder why our result does not include the term
+3
+|V | in the lower bound for the
+average degree as Conjecture 1.1 states. The reason is that we can construct some infinite
+families of graphs with maximum degree 7 and average degree 6 which satisfy all currently
+known adjacency lemmas. For example, for any positive integer t, consider a graph G with
+degree sequence (4t, 72t) such that each 4-vertex is adjacent to four 7-vertices and each 7-vertex
+is adjacent two 4-vertices. One can easily check that G satisfies all adjacent lemmas that we
+currently have and d(G) = 7 − 1 = 6. The above example can be generalized for arbitrary
+maximum degree ∆ = 2k + 1 ≥ 7. For each t ≥ 1, let G be a graph with degree sequence
+(kt, ∆kt) such that each k-vertex is adjacent to k vertices of degree ∆ and each ∆-vertex is
+adjacent to exactly one k-vertex. Then d(G) = ∆ − 1 = 2k and G satisfies all adjacency
+lemmas that we know.
+The above examples and several other examples not only present a challenge but also
+indicate the necessity to develop new adjacency lemmas to attack Conjecture 1.1 and other
+edge coloring problems. In particular, so far all adjacency lemmas are about a path on at
+most four vertices. Lemma 2.9 is indeed a lemma that deals with a path with five vertices
+and it is the key lemma in the proof of our main result, but it is only for degree 3-vertices.
+To completely solve the case of 7-critical graphs and beyond, more general adjacency lemmas
+concerning paths on five vertices are needed although it is very challenging to develop such
+lemmas. It would be practical and very useful to use computer program to complete the
+remaining cases for 7-critical graphs and to develop some forbidden structures for critical
+graphs in general.
+4
+Applications to graphs embedded on surfaces with nonneg-
+ative Euler characteristics
+Theorem 3.1 clearly implies that every planar graph with maximum degree 7 is class one
+which was conjectured by Vizing and independently proved by Sanders and Zhao [17], and
+Zhang [23] and its extension to projective planar graphs [18] since every graph which can be
+embedded in a plane or a projective plane has average degree strictly less than 6. Our result
+also implies the following result due to Sanders and Zhao [18].
+Theorem 4.1 (Sanders and Zhao [18]) Let G be a graph with maximum degree 7. If G can
+be embedded in the torus or Klein bottle, then G is class one.
+Proof.
+Prove by contradiction.
+Suppose that G is not class one.
+Then we may assume
+that G is 7-critical. By Euler’s formula, d(G) ≤ 6. By Theorem 3.1, we have d(G) = 6
+and d(f) = 3 for each face f. Since G is simple, we further have δ ≥ 3. Denote by M′(x)
+the new charge of the vertex x and A, B, C the sets defined in the previous section. Then
+�
+x∈V (G) M′(x) = �
+x∈V (G)(d(G) − 6) = 0. Thus M′(x) = 0 for every vertex x in G.
+Since δ(G) ≥ 3, we have A = ∅. By (II) and (IV) in the proof of Theorem 3.1, d(u)+d(v) ≥
+∆ + 3 and there are no 5-vertices in G. Thus B = ∅. Since every face is a 3-face and G is
+2-connected, every two adjacent vertices share at least two common neighbors.
+7
+
+Claim 4.1.1 δ(G) = 4 and every 4-vertex is adjacent to exactly two 7-vertices and two 6-
+vertices.
+Proof. Let y be a 7-vertex with a neighbor x where 3 ≤ d(x) ≤ 4. Since any two adjacent
+vertices share at least two neighbors, by Lemma 2.6, y is adjacent to only one 4−-vertex.
+Since there are no 5-vertices in G, y is adjacent to exactly one 5−-vertex. This implies C = ∅.
+Therefore A = B = C = ∅.
+Hence every 7-vertex is adjacent to a 4−-vertex otherwise
+M′(x) = M(x) = 1 > 0 if x is a 7-vertex without a 4−-neighbor. Therefore every 7-vertex has
+exactly one 4−-neighbor.
+If there is a 3-vertex, by Lemma 2.3, there is one 7-vertex x that has no 4−-neighbors,
+a contradiction. Therefore δ = 4 and every 7-vertex is adjacent to exactly one 4-vertex. By
+(III), every 4-vertex is adjacent to exactly two 7-vertices and two 6-vertices.
+Denote by Vi the set of i-vertices and ni = |Vi|. Then by Claim 4.1.1, n4 = 2n7 and
+n4 ≤ 2n6.
+Since every 7-vertex is adjacent to a 4-vertex, every 7-vertex is adjacent to at least 4 vertices
+in V7 and every vertex has at least two neighbors in V7 by Lemma 2.1. Thus 2n6 + 2n4 ≤
+|[V7, V6 ∪ V4]| ≤ 3n7. This implies 6n7 = 3n4 ≤ 3n7. This contradiction completes the proof
+of the theorem.
+5
+Proofs of new lemmas
+Before giving the proofs, we first introduce some notations and lemmas that are needed in
+this section.
+The set of all k-edge-colorings of a graph G is denoted by Ck(G). Let ϕ ∈ Ck(G). For any
+color α, let Eα = {e ∈ E : ϕ(e) = α}. For any two distinct colors α and β, denote by Gϕ(α, β)
+the subgraph of G induced by Eα ∪ Eβ. The components of Gϕ(α, β) are called (α, β)-chains.
+Clearly, each (α, β)-chain is either a path or a cycle of edges alternately colored with α and β.
+For each (α, β)-chain P, let ϕ/P denote the k-edge-coloring obtained from ϕ by exchanging
+colors α and β on P.
+For any v ∈ V , let Pv(α, β, ϕ) denote the unique (α, β)-chain containing v. Notice that,
+for any two vertices u, v ∈ V , either Pu(α, β, ϕ) = Pv(α, β, ϕ) or Pu(α, β, ϕ) is vertex-disjoint
+from Pv(α, β, ϕ). This fact will be used very often without mentioning. For convenience, we
+define Pv(α, β, ϕ) = v and ϕ/Pv(α, β, ϕ) = ϕ when α = β.
+For any v ∈ V , let ϕ(v) = {ϕ(e) : e ∈ E(v)} denote the set of colors presented at v and
+¯ϕ(v) = C \ ϕ(v) the set of colors not assigned to any edge incident to v, which are called
+missing colors at v. For a vertex set X ⊆ V (G), we call X elementary (with respect to ϕ) if
+all missing color sets ¯ϕ(x) (x ∈ X) are mutually disjoint.
+A multi-fan at x with respect to the edge e = xy ∈ E(G) and the coloring ϕ ∈ C∆(G − e)
+is a sequence F = (x, e1, y1, . . . , ep, yp) with p ≥ 1 consisting of edges e1, e2, . . . , ep and vertices
+x, y1, y2, . . . , yp satisfying the following two conditions:
+• The edges e1, e2, . . . , ep are distinct, e1 = e and ei = xyi for i = 1, . . . , p.
+8
+
+• For every edge ei with 2 ≤ i ≤ p, there is a vertex yj with 1 ≤ j < i such that
+ϕ(ei) ∈ ¯ϕ(yj).
+Note that a multi-fan is slightly more general than a Vizing-fan which requires j = i − 1
+in the second condition.
+Lemma 5.1 (Stiebitz, Scheide, Toft and Favrholdt [19]) Let G be a ∆-critical graph, xy1 =
+e ∈ E(G) and ϕ ∈ C∆(G − e). If F = (x, e1, y1, . . . , ep, yp) is a multi-fan at x with respect to
+e and ϕ. Then the following statements hold:
+(a) {x, y1, y2, . . . , yp} is elementary.
+(b) If α ∈ ¯ϕ(x) and β ∈ ¯ϕ(yi) for some i, then Px(α, β, ϕ) = Pyi(α, β, ϕ).
+The following lemma is a direct corollary of Lemma 5.1.
+Lemma 5.2 Let G be a ∆-critical graph, xy = e ∈ E(G) and ϕ ∈ C∆(G − e). Let xyz be a
+path.
+(1) If d(z) ≤ 2∆ − (d(x) + d(y)) + 1, then α = ϕ(yz) ∈ ϕ(x) ∩ ϕ(y) and for any color
+β ∈ ¯ϕ(z) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)), Pz(α, β, ϕ) ends at x or y.
+(2) If ϕ(yz) ∈ ¯ϕ(x), then ¯ϕ(x) ∪ ¯ϕ(y) ⊆ ϕ(z) and thus d(z) ≥ 2∆ − (d(x) + d(y)) + 2.
+A Kierstead path with respect to e = y0y1 and ϕ ∈ C∆(G − e) is a path K = y0y1 · · · yp
+with p ≥ 1 such that for every edge yiyi+1 with 1 ≤ i ≤ p − 1, there is a vertex yj with
+0 ≤ j < i such that ϕ(yiyi+1) ∈ ¯ϕ(yj).
+Clearly a Kierstead path with 3 vertices is a multi-fan with center y1.
+The next two
+lemmas are elementary properties of a Kierstead path with 4 vertices.
+Lemma 5.3 (Kostochka and Stiebitz [19], Luo and Zhao [15]) Let G be a ∆-critical graph,
+y0y1 = e ∈ E(G) and ϕ ∈ C∆(G − e). Let K = y0y1y2y3 be a Kierstead path with respect
+to e and ϕ.
+Then V (K) is elementary unless d(y1) = d(y2) = ∆(G), in which case, all
+colors in ¯ϕ(y0), ¯ϕ(y1), ¯ϕ(y2) and ¯ϕ(y3) are distinct except one possible common missing color
+in ¯ϕ(y3) ∩ ( ¯ϕ(y0) ∪ ¯ϕ(y1)).
+Lemma 5.4 Let G be a ∆-critical graph, y0y1 = e ∈ E(G) and ϕ ∈ C∆(G − e). Suppose that
+K = y0y1y2y3 is a Kierstead path with respect to e and ϕ, min{d(y1), d(y2)} < ∆, α ∈ ¯ϕ(y3)
+and β ∈ ¯ϕ(yi) for some i ∈ {0, 1, 2}. If β /∈ {ϕ(y1y2), ϕ(y2y3)}, then Py3(α, β, ϕ) ends at yi.
+Proof. Since K is a Kierstead path and {y0, y1, y2, y3} is elementary by Lemma 5.3, we have
+α /∈ {ϕ(y1y2), ϕ(y2y3)}. Suppose to the contrary that Py3(α, β, ϕ) does not end at yi. Then
+after interchanging α, β on this path, K is still a Kierstead path, but β is missing at both yi
+and y3, a contradiction to Lemma 5.3. This completes the proof.
+A ϕ-broom (Figure 1 (a)) with respect to y0y1 and ϕ ∈ C∆(G − y0y1) is a sequence
+B = (y0, e1, y1, . . . , ep, yp) with p ≥ 3 such that e1 = y0y1, e2 = y1y2, ϕ(e2) ∈ ¯ϕ(y0) and for
+all i ≥ 3, ei = y2yi and ϕ(ei) ∈ ¯ϕ(yj) for some j < i.
+Lemma 5.5 (Cao, Chen, Jing, Stiebitz and Toft [7]) Let G be a ∆-critical graph, y0y1 = e1 ∈
+E(G) and ϕ ∈ C∆(G−e1). If B = (y0, e1, y1, . . . , ep, yp) is a ϕ-broom and min{d(y1), d(y2)} <
+∆, then the vertex set of B is elementary.
+9
+
+…
+0
+y
+1
+y
+2
+y
+3
+y
+4
+y
+p
+y
+
+
+.
+ 
+some
+for 
+ )
+(
+)
+(
+ 
+Broom.
+ )
+(
+2
+i
+j
+y
+y
+y
+a
+j
+i
+
+
+
+2
+
+u
+b
+a
+1t
+2t
+1s
+2s
+c
+ 
+Kite.
+ )
+(b
+
+
+1
+
+2
+
+2
+
+u
+b
+a
+1t
+2t
+1s
+2s
+1
+
+1
+
+.)
+(
+)
+(
+,
+,
+,
+ 
+Fork.
+ )
+(
+2
+1
+2
+1
+b
+a
+c
+
+
+
+
+
+
+
+
+Figure 1: Brooms, kites and forks.
+A kite H (Figure 1 (b)) is a graph with
+V (H) = {a, b, c, u, s1, s2, t1, t2} and E(H) = {ab, ac, bu, cu, us1, us2, s1t1, s2t2}.
+The lemma below reveals some properties of a kite with specified colors on its edges.
+Lemma 5.6 (Cao, Chen and Shan [8]) Let G be a ∆-critical graph, H ⊆ G be a kite with
+V (H) = {a, b, c, u, s1, s2, t1, t2}, and let ϕ ∈ C∆(G − ab). Suppose that both K = abus1t1
+and K∗ = bacus2t2 are Kierstead paths with respect to ab and ϕ. If ϕ(s1t1) = ϕ(s2t2), then
+| ¯ϕ(t1) ∩ ¯ϕ(t2) ∩ ( ¯ϕ(a) ∪ ¯ϕ(b))| ≤ 4.
+Let G be a ∆-critical graph, ab ∈ E(G), and ϕ ∈ C∆(G−ab). A fork H (Figure 1 (c)) with
+respect to ϕ is a graph with V (H) = {a, b, u, s1, s2, t1, t2} and E(H) = {ab, bu, us1, us2, s1t1, s2t2}
+such that ϕ(bu) ∈ ¯ϕ(a), ϕ(us1), ϕ(us2) ∈ ¯ϕ(a) ∪ ¯ϕ(b), and ϕ(s1t1) ∈ ( ¯ϕ(a) ∪ ¯ϕ(b)) ∩ ¯ϕ(t2) and
+ϕ(s2t2) ∈ ( ¯ϕ(a) ∪ ¯ϕ(b)) ∩ ¯ϕ(t1). Forks may not exist in a ∆-critical graph if the degree sum
+of a, t1 and t2 is small.
+Lemma 5.7 (Cao and Chen [6]) Let G be a ∆-critical graph, ab ∈ E(G), and {u, s1, s2, t1, t2} ⊆
+V (G). If ∆ ≥ dG(a) + dG(t1) + dG(t2) + 1, then for any ϕ ∈ C∆(G − ab), G does not contain
+a fork on {a, b, u, s1, s2, t1, t2} with respect to ϕ.
+5.1
+Proof of Lemma 2.7
+Lemma 2.7 Let G be a ∆-critical graph with ∆ ≥ 7. Then |N∆∼2(v)| ≤ 5 for every v ∈ V (G).
+10
+
+Proof. Suppose to the contrary that there is a ∆-vertex v with |N∆∼2(v)| ≥ 6. By Lemma 2.4,
+v has no 2-neighbors and by Lemma 2.1, each vertex z ∈ N∆∼2(v) has exactly one 2-neighbor.
+Let N2(v) = N(N(v))\N[v]. Since |N∆∼2(v)| ≥ 6, there are at least three 2-vertices in N2(v).
+Let x be a 2-vertex in N2(v) and y be a vertex in N(x) ∩ N(v). Clearly y ∈ N∆∼2(v). Let
+ϕ ∈ C∆(G − xy). Then ¯ϕ(x) ∪ ¯ϕ(y) = C. We first point out one fact that will be used very
+often.
+Fact 1. Let t1, t2 be two 2-vertices in N2(v)\{x}, s1 ∈ N(v) ∩ N(t1) and s2 ∈ N(v) ∩ N(t2).
+(a) If |N(x) ∩ N(v)| = 2 and ϕ(s1t1) = ϕ(s2t2), then ϕ(t1) ̸= ϕ(t2).
+(b) If ϕ(s1t1) ̸= ϕ(s2t2), then either ϕ(s1t1) ∈ ϕ(t2) or ϕ(s2t2) ∈ ϕ(t1).
+Proof. (a) Denote N(x) ∩ N(v) = {y, z}. Suppose to the contrary that ϕ(t1) = ϕ(t2). Then
+| ¯ϕ(t1)∩ ¯ϕ(t2)| ≥ 5 since ∆ ≥ 7, and {x, y, z, v, s1, s2, t1, t2} form a kite with ϕ(s1t1) = ϕ(s2t2),
+a contradiction to Lemma 5.6.
+(b) Suppose to the contrary that ϕ(s1t1) ∈ ¯ϕ(t2) and ϕ(s2t2) ∈ ¯ϕ(t1). Then {x, y, v, s1, s2, t1, t2}
+form a fork with ∆ ≥ 7 = d(x) + d(t1) + d(t2) + 1, a contradiction to Lemma 5.7.
+We consider two cases in the following: there are three 2-vertices in N2(v), or there are at
+least four 2-vertices in N2(v).
+Case 1: There are exactly three 2-vertices in N2(v).
+Let t1, t2 be the two 2-vertices in N2(v)\{x}. Since N∆∼2(v) ≥ 6, we have |N(ti)∩N(v)| =
+2 for each i = 1, 2 and |N(x) ∩ N(v)| = 2. Let N(ti) ∩ N(v) = {si, s′
+i} for each i = 1, 2. By
+the symmetry between si and s′
+i, we may assume that ϕ(s1t1) ̸= ϕ(s2t2). By Fact 1(b), we
+may assume ϕ(s′
+1t1) = ϕ(s2t2). Applying Fact 1(a) on s′
+1, t1, s2, t2, we have ϕ(t1) ̸= ϕ(t2).
+Thus ϕ(s′
+2t2) ̸= ϕ(s1t1), ϕ(s′
+2t2) ̸∈ ϕ(t1) and ϕ(s1t1) /∈ ϕ(t2). This gives a contradiction to
+Fact 1(b) on s1, t1, s′
+2, t2.
+Case 2: There are at least four 2-vertices in N2(v).
+Let t1, t2, t3 be three 2-vertices in N2(v)\{x}, si be a vertex in N(ti) ∩ N(v), and s′
+i be
+the other neighbor of ti for each i = 1, 2, 3.
+Claim A. ϕ(siti) ̸= ϕ(sjtj) for any 1 ≤ i < j ≤ 3.
+Proof. Prove by contradiction. Since ∆ ≥ 7 > d(t1)+d(t2)+d(t3), let η ∈ ¯ϕ(t1)∩ ¯ϕ(t2)∩ ¯ϕ(t3).
+By symmetry, we only need to consider the following two cases: ϕ(s1t1) = ϕ(s2t2) = ϕ(s3t3) =
+α, or ϕ(s1t1) = ϕ(s2t2) = α and ϕ(s3t3) = β ̸= α.
+Suppose that ϕ(s1t1) = ϕ(s2t2) = ϕ(s3t3) = α. Then by symmetry, we may assume that
+Pt1(α, η, ϕ) does not pass through t2, t3. Let ϕ′ = ϕ/Pt1(α, η, ϕ). Then s1, t1, s2, t2 give a
+contradiction to Fact 1(b) under ϕ′.
+Suppose that ϕ(s1t1) = ϕ(s2t2) = α and ϕ(s3t3) = β ̸= α. If Pt1(α, η, ϕ) does not end
+at t2, let ϕ′ = ϕ/Pt1(α, η, ϕ).
+Then s1, t1, s2, t2 give a contradiction to Fact 1 (b) under
+ϕ′.
+Thus Pt1(α, η, ϕ) ends at t2, so Pt3(α, η, ϕ) does not pass through t1, t2.
+Let ϕ1 =
+ϕ/Pt3(α, η, ϕ). Now α ∈ ¯ϕ1(t3). Then by Fact 1(b), we have ϕ1(t1) = ϕ1(t2) = {α, β}. Let
+η′ ∈ ¯ϕ1(t1) ∩ ¯ϕ1(t2) ∩ ¯ϕ(t3). By symmetry, we may assume that Pt3(β, η′, ϕ1) does not pass
+11
+
+through t1. Let ϕ2 = ϕ1/Pt3(β, η′, ϕ1). Then s1, t1, s3, t3 give a contradiction to Fact 1(b)
+under ϕ2. This proves Claim A.
+Let ϕ(s1t1) = α, ϕ(s2t2) = β, ϕ(s3t3) = γ.
+Claim B. {ϕ(s′
+1t1), ϕ(s′
+2t2), ϕ(s′
+3t3)} = {ϕ(s1t1), ϕ(s2t2), ϕ(s3t3)}.
+Proof. By Claim A, α, β, γ are distinct. Suppose that ϕ(t1) = {α, η} where η /∈ {β, γ}. By
+Fact 1(b), we have ϕ(t2) = {β, α} and ϕ(t3) = {γ, α}. Then s2, t2, s3, t3 give a contradiction
+to Fact 1(b). Thus by symmetry, we may assume that ϕ(t1) = {α, β}. Now by applying Fact
+1(b) on s1, t1, s3, t3, we have ϕ(t3) = {α, γ}; By applying Fact 1(b) on s2, t2, s3, t3, we have
+ϕ(t2) = {β, γ}. This proves Claim B.
+The final step. Without loss of generality, assume ϕ(t1) = {α, β}. Since |N∆∼2| ≥ 6, let
+s4 ∈ N∆∼2\{s1, s2, s3} and t4 be the 2-neighbor of s4. If t4 ∈ {t1, t2, t3}, then by symmetry,
+we may assume that t4 = t1.
+Then ϕ(s4t1) = β and s4, t1, s3, t3 give a contradiction to
+Fact 1(b). If t4 ̸∈ {t1, t2, t3}, then by Claim A, ϕ(s4t4) ̸= ϕ(siti) for each i = 1, 2, 3. Thus
+{s1, s2, s4, t1, t2, t4} does not satisfy Claim B. This completes the proof of Case 2 and thus of
+Lemma 2.7.
+5.2
+Proof of Lemma 2.8
+Lemma 2.8 Let G be a ∆-critical graph and xyrst be a path with d(x) + d(y) = ∆ + 2 and
+max{d(x), d(y)} < ∆. Then d(t) ≥ ∆ − 2.
+Proof. Let ϕ ∈ C∆(G − xy). Since d(x) + d(y) = ∆ + 2, we have ¯ϕ(x) ∪ ¯ϕ(y) = C. Let
+ϕ(yr) = α, ϕ(rs) = β, ϕ(st) = γ. Then α ∈ ¯ϕ(x) and β, γ ∈ ¯ϕ(x) ∪ ¯ϕ(y). Since d(x) < ∆ and
+d(y) < ∆, we have | ¯ϕ(x)| ≥ 2 and | ¯ϕ(y)| ≥ 2. Suppose to the contrary that d(t) ≤ ∆ − 3.
+Then | ¯ϕ(t)| ≥ 3.
+Claim A. There is a coloring in C∆(G − xy) such that yr and st are colored differently, i.e.,
+we may assume α ̸= γ.
+Proof. Suppose to the contrary that α = γ. Since d(t) ≤ ∆ − 3, let η ∈ ¯ϕ(t) \ {α, β}.
+If η ∈ ¯ϕ(y), then Px(α, η, ϕ) = Py(α, η, ϕ) by Lemma 5.1 and thus is disjoint from
+Pt(α, η, ϕ). Let ϕ1 = ϕ/Pt(α, η, ϕ). Then ϕ1(yr) ̸= ϕ1(st), as desired.
+Suppose η ∈ ¯ϕ(x). Since | ¯ϕ(y)| ≥ 2, let δ ∈ ¯ϕ(y) \ {β}. Clearly δ ̸∈ {ϕ(yr), ϕ(rs), ϕ(st)}.
+Let ϕ1 = ϕ/Px(δ, η, ϕ) and we are back to the case when η ∈ ¯ϕ(y). This proves Claim A.
+From now on, we assume that α ̸= γ in the following proof.
+Claim B. We may further assume that α, β ∈ ¯ϕ(t).
+Proof. We consider two cases: β ∈ ¯ϕ(t) and β /∈ ¯ϕ(t).
+Case B.1: β ∈ ¯ϕ(t).
+We may assume α ∈ ϕ(t) otherwise we are done. Let η ∈ ¯ϕ(t) \ {α, β}. Clearly η ̸= γ
+since ϕ(st) = γ.
+If η ∈ ¯ϕ(y), let ϕ1 = ϕ/Pt(α, η, ϕ). Then we have α, β ∈ ¯ϕ1(t), as desired.
+12
+
+If η ∈ ¯ϕ(x), let δ ∈ ¯ϕ(y) \ {β}. By Lemma 5.1, regardless of whether δ = γ or not,
+Px(δ, η, ϕ) does not contain yr, rs or st since η ∈ ¯ϕ(t). Let ϕ1 = ϕ/Px(δ, η, ϕ) and we are
+back to the case when η ∈ ¯ϕ(y). This completes the proof of Case B.1.
+Case B.2: β /∈ ¯ϕ(t).
+Case B.2.1: α ∈ ¯ϕ(t).
+If β ∈ ¯ϕ(y), then by Lemma 5.1, Px(α, β, ϕ) is disjoint from Pt(α, β, ϕ). Thus Pt(α, β, ϕ)
+does not contain yr or rs. Let ϕ1 = ϕ/Pt(α, β, ϕ). Then β ∈ ¯ϕ1(t) and we are back to Case
+B.1.
+Now assume β ∈ ¯ϕ(x). If there is a color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t), let ϕ1 = ϕ. Otherwise, let δ ∈
+¯ϕ(y) and η ∈ ¯ϕ(t)\{α}. Then Pt(η, δ, ϕ) does not pass through x or y. Let ϕ1 = ϕ/Pt(η, δ, ϕ).
+Then δ ∈ ¯ϕ1(y) ∩ ¯ϕ1(t) and β ∈ ¯ϕ1(x). Note that Px(δ, β, ϕ1) and Pt(δ, β, ϕ1) are disjoint. If
+Pt(δ, β, ϕ1) does not contain rs, let φ2 = ϕ1/Pt(δ, β, ϕ1) and then ϕ2 is a desired coloring. If
+Px(δ, β, ϕ1) does not contain rs, let φ2 = ϕ1/Px(δ, β, ϕ1). Then β ∈ ¯ϕ2(y) and we are back to
+the case when β ∈ ¯ϕ(y). This proves Case B.2.1.
+Case B.2.2: α /∈ ¯ϕ(t).
+If there is a color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t), let ϕ1 = ϕ/Pt(α, δ, ϕ). Then α ∈ ¯ϕ1(t) and we are
+back to Case B.2.1.
+Suppose ¯ϕ(y)∩ ¯ϕ(t) = ∅. Let η ∈ ¯ϕ(t) and δ ∈ ¯ϕ(y)\{β}. Then δ ∈ ϕ(t). By Lemma 5.1,
+regardless of whether δ = γ or not, Px(δ, η, ϕ) does not contain yr, rs or st since η ∈ ¯ϕ(t). Let
+ϕ1 = ϕ/Px(δ, η, ϕ), we are back to the case when ¯ϕ(y) ∩ ¯ϕ(t) ̸= ∅. This completes the proof
+of Case B.2 and thus the proof of Claim B.
+By Claim B, we assume that α, β ∈ ¯ϕ(t) in the following proof.
+Claim C. We may further assume that β, γ ∈ ¯ϕ(y).
+Proof. We consider two cases: β ∈ ¯ϕ(y) and β /∈ ¯ϕ(y).
+Case C.1: β ∈ ¯ϕ(y).
+We may assume γ ∈ ϕ(y) otherwise we are done. Let η ∈ ¯ϕ(t) \ {α, β}.
+Similar to the argument in Case B.2, we may assume that there is a color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t)
+and δ ̸= β. Then Pt(δ, γ, ϕ) and Px(δ, γ, ϕ) are disjoint. Let ϕ1 = ϕ/Px(δ, γ, ϕ). Then we
+have β, γ ∈ ¯ϕ1(y), as desired. This completes the proof of Case C.1.
+Case C.2: β /∈ ¯ϕ(y).
+If γ ∈ ¯ϕ(y), then Pt(γ, β, ϕ) and Px(γ, β, ϕ) are disjoint by Lemma 5.1. Note that rs and
+st are contained in Pt(γ, β, ϕ). Let ϕ1 = ϕ/Px(γ, β, ϕ). Then β ∈ ¯ϕ1(y) and we are back to
+Case C.1.
+Suppose γ ∈ ¯ϕ(x). Similar to the argument in Case B.2, we can assume that there is a
+color δ ∈ ¯ϕ(y) ∩ ¯ϕ(t). Then δ ̸∈ {α, β}. Thus Px(η, γ, ϕ) is disjoint from Pt(η, γ, ϕ), so it
+does not contain st since η ∈ ¯ϕ(t). Let ϕ1 = ϕ/Px(η, γ, ϕ) and we are back to the case when
+γ ∈ ¯ϕ(y). This completes the proof of Case C.2, and thus Claim C holds.
+Now by Claims A, B and C, we assume that ϕ ∈ C∆(G − xy) satisfies the following
+properties:
+13
+
+• ϕ(yr) = α, ϕ(rs) = β, ϕ(st) = γ,
+• α ̸= γ,
+• α, β ∈ ¯ϕ(t) and β, γ ∈ ¯ϕ(y).
+Let ϕ1 = ϕ/Pt(α, γ, ϕ). Under the coloring ϕ1, Py(β, α, ϕ1) = yrst ends at t but not x, a
+contradiction to Lemma 5.1. This completes the proof of Lemma 2.8.
+5.3
+Proof of Lemma 2.9
+Lemma 2.9 Let G be a ∆-critical graph and xyrst be a path with d(x) = 3 and d(y) = ∆.
+Suppose that y has a neighbor z ̸∈ {x, r, s} with d(z) ≤ ∆ − 2. Then d(s) ≥ ∆ − 1; and
+d(z) + d(t) ≥ ∆ + 1 if d(t) ≤ ∆ − 4.
+Proof. Let ϕ be a coloring in C∆(G − xy). Since d(z) ≤ ∆ − 2, d(x) = 3 and d(y) = ∆,
+we have |ϕ(x) ∩ ϕ(y)| = 1. By Lemma 5.2, without loss of generality, assume ϕ(x) = {1, 2},
+ϕ(yz) = 2, ϕ(yr) = 3. Denote ϕ(rs) = β and ϕ(st) = γ. Note that ¯ϕ(y) = {1}.
+(1) We first show d(s) ≥ ∆ − 1.
+Suppose to the contrary d(s) ≤ ∆ − 2.
+We first consider the case when ϕ(rs) = β ̸= 2 = ϕ(yz). Then K = xyrs is a Kierstead
+path. By Lemma 5.3, | ¯ϕ(s) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)| ≤ 1. Thus d(s) ≥ 2∆ − (d(x) + d(y) + 1 = ∆ − 2.
+Since d(s) ≤ ∆ − 2, we have d(s) = ∆ − 2. Note that d(s) = ∆ − 2 only if 2 ∈ ¯ϕ(s) and
+| ¯ϕ(s) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y))| = 1. Denote ¯ϕ(s) = {2, α}.
+If ¯ϕ(z) \ {α, β} ̸= ∅, then η ∈ ¯ϕ(z) \ {α, β}. By Lemma 5.2, Pz(η, 2, ϕ) ends at x or y
+and thus it does not pass through s. Let ϕ1 = ϕ/Pz(η, 2, ϕ). Then xyrs remains a Kierstead
+path with respect to ϕ1 and xy. However, ¯ϕ1(s) = {2, α} ⊆ ¯ϕ(x) ∪ ¯ϕ(y), a contradiction to
+Lemma 5.3. Therefore ¯ϕ(z) \ {α, β} = ∅. Since d(z) ≤ ∆ − 2, we have ¯ϕ(z) = {α, β}.
+If β ̸= 1, then we may assume α = 1. Otherwise both Pz(1, α, ϕ) and Ps(1, α, ϕ) are disjoint
+from Px(1, α, ϕ). Let ϕ2 = ϕ/(Pz(1, α, ϕ) ∪ Ps(1, α, ϕ)). Then 1 is missing at both z and s
+and 3, β ∈ ¯ϕ1(x) ∪ ¯ϕ1(y). Since 1 ∈ ¯ϕ(z) ∩ ¯ϕ(s), both Pz(1, 3, ϕ) and Ps(1, 3, ϕ) are disjoint
+from Px(1, 3, ϕ) and thus neither passes through x, y. Let ϕ2 = ϕ/(Pz(1, 3, ϕ) ∪ Ps(1, 3, ϕ)).
+Then 3 ∈ ¯ϕ2(z) ∩ ¯ϕ2(s) and 2 ∈ ϕ2(x) ∩ ϕ2(y). By Lemma 5.2, Pz(2, β, ϕ2) ends at either x or
+y and thus is disjoint from Ps(2, β, ϕ2). Let ϕ3 = ϕ2/Ps(2, β, ϕ2). Then Pz(2, β, ϕ3) = zyrs
+which does not end at x or y, a contradiction to Lemma 5.2.
+Now assume β = 1. Then ¯ϕ(z) = {1, α} and thus Ps(1, α, ϕ) does not pass through x, y, z.
+Interchange colors on Ps(1, α, ϕ) and we are back to the case when β ̸= 1. Therefore this
+completes the proof when β ̸= ϕ(yz).
+Now we consider the case when β = ϕ(yz) = 2. Let η be a color in ¯ϕ(z). Clearly η ̸= 2.
+If η = 1, then by recoloring yz with 1, we are back to the case when β ̸= ϕ(yz).
+Thus
+η ∈ ¯ϕ(x). Then Px(η, 1, ϕ) = Py(η, 1, ϕ). Thus by interchanging η and 1 on Px(η, 1, ϕ) and
+then recoloring yz with η, we are back to the case when β ̸= ϕ(yz). This completes the proof
+that d(s) ≥ ∆ − 1.
+(2) Now we assume d(t) ≤ ∆ − 4 and show d(z) + d(t) ≥ ∆ + 1.
+14
+
+Suppose to the contrary that d(z) + d(t) ≤ ∆.
+Claim A. There is a coloring in C∆(G − xy) such that yr and st receive distinct colors,
+i.e., we may assume that γ ̸= 3.
+Proof. Suppose to the contrary that γ = 3. Let η ∈ ¯ϕ(t) \ {2, 3, β}. Then η ∈ ¯ϕ(x) ∩ ¯ϕ(y).
+If η = 1, then Px(3, η, ϕ) = Py(3, η, ϕ) by Lemma 5.1, so Pt(3, η, ϕ) is disjoint from
+Px(3, η, ϕ). Let ϕ1 = ϕ/Pt(3, η, ϕ). We have that ϕ1(yr) ̸= ϕ1(st) now.
+If η ̸= 1, then Pt(1, η, ϕ) does not contain x or y. Let ϕ1 = ϕ/Pt(1, η, ϕ) and we are back
+to the previous case. This proves Claim A.
+From now on, we assume that ϕ(yr) ̸= ϕ(st) (i.e. γ ̸= 3) in the following proof.
+Claim B. We may further assume that 3, β ∈ ¯ϕ(t).
+Proof. We split the proof into two cases: β ∈ ¯ϕ(t) and β /∈ ¯ϕ(t).
+Case B.1: ϕ(rs) = β ∈ ¯ϕ(t).
+Case B.1.1: β ̸∈ ¯ϕ(y). Then β ̸= 1.
+If 1 ∈ ¯ϕ(t), then Pt(1, 3, ϕ) is disjoint from Px(1, 3, ϕ) = Py(1, 3, ϕ) and yr, rs ̸∈
+Pt(1, 3, ϕ). Let ϕ1 = ϕ/Pt(1, 3, ϕ). Then ϕ1(yr) = 3, ϕ1(rs) = β, ϕ1(st) = γ and 3, β ∈ ¯ϕ1(t),
+as desired.
+Now assume 1 ̸∈ ¯ϕ(t). Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t) \ {2, 3, β}. Then η ∈ ¯ϕ(x).
+Thus Pt(1, η, ϕ) does not pass through x or y and does not contain yr, rs, or st. Let ϕ1 =
+ϕ/Pt(1, η, ϕ) and we are back to the case when 1 ∈ ¯ϕ(t).
+Case B.1.2: β ∈ ¯ϕ(y). Then β = 1.
+If γ ̸= 2, then γ ∈ ¯ϕ(x).
+Thus, Pt(1, γ, ϕ) is disjoint from Px(1, γ, ϕ).
+Let ϕ1 =
+ϕ/Pt(1, γ, ϕ). Then ϕ1(rs) = γ ∈ ¯ϕ1(x) ∩ ¯ϕ1(t) and γ ̸= 1. We are back to Case B.1.1.
+Now assume ϕ(st) = γ = 2. Since d(z) + d(t) ≤ ∆ and 2 ∈ ϕ(z) ∩ ϕ(t), there is a color
+η ∈ ¯ϕ(z) ∩ ¯ϕ(t). Since η ̸= γ, we have η ∈ ¯ϕ(x) ∪ ¯ϕ(y).
+If η ̸= 1, by Lemma 5.2, Pz(2, η, ϕ) ends at x. Thus Pt(2, η, ϕ) does not pass through x or
+y and does not contain the edge rs. Let ϕ1 = ϕ/Pt(2, η, ϕ). Then ϕ1(st) = η ∈ ¯ϕ1(x) ∪ ¯ϕ1(y)
+and we are back to the previous case
+If η = 1, then Pz(1, 2, ϕ) = yz. Let ϕ1 = ϕ/Pz(1, 2, ϕ) and we are back to Case B.1.1.
+This completes the proof of Case B.1.
+Case B.2: ϕ(rs) = β /∈ ¯ϕ(t).
+Case B.2.1: ϕ(yr) = 3 ∈ ¯ϕ(t).
+Case B.2.1.1: β ∈ ¯ϕ(y). That is β = 1.
+Then Px(3, β, ϕ) ends at y by Lemma 5.1 and it contains both yr and rs.
+Thus
+Px(3, β, ϕ) and Pt(3, β, ϕ) are disjoint. Let ϕ1 = ϕ/Pt(3, β, ϕ). Then β ∈ ¯ϕ1(t) and we are
+back to Case B.1.
+Case B.2.1.2: β = ϕ(yz) = 2.
+If 1 ∈ ¯ϕ(z), recolor yz with 1. We are back to Case B.2.1.1.
+Assume 1 ̸∈ ¯ϕ(z). Since d(z) ≤ ∆ − 2 and 2 ∈ ϕ(z), let η ∈ ¯ϕ(z)\{3}. Clearly η ̸= 1, 2
+and η ∈ ¯ϕ(x). Then Pz(1, η, ϕ) does not pass through x or y and does not contain the edge
+15
+
+rs. Let ϕ1 = ϕ/Pz(1, η, ϕ). Then 1 ∈ ¯ϕ1(z) and we are back to the previous case.
+Case B.2.1.3: β ∈ ¯ϕ(x).
+We may further assume 1 ∈ ¯ϕ(t). Otherwise, since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t) \ {2, 3}.
+Then η ̸∈ {1, 2, 3, β}, and Px(1, η, ϕ) and Pt(1, η, ϕ) are disjoint. Let ϕ1 = ϕ/Pt(1, η, ϕ). Then
+1 ∈ ¯ϕ1(t).
+Note Px(β, 1, ϕ1) and Pt(β, 1, ϕ1) are disjoint. If Px(β, 1, ϕ1) does not contain the edge
+rs, let ϕ2 = ϕ1/Px(β, 1, ϕ1) and we are back to Case B.2.1.1. If Pt(β, 1, ϕ1) does not contain
+the edge rs, let ϕ2 = ϕ1/Pt(β, 1, ϕ1) and we are back to Case B.1. This completes the proof
+of Case B.2.1.
+Case B.2.2: 3 /∈ ¯ϕ(t).
+Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t) \ {2, 3, β}.
+If η = 1, then Px(3, 1, ϕ) and Pt(3, 1, ϕ) are disjoint. Let ϕ1 = ϕ/Pt(1, 3, ϕ). Then
+ϕ1(yr) = 3 ∈ ¯ϕ1(t) and we are back to Case B.2.1.
+Therefore η ̸= 1. If β ̸= 1, then Px(1, η, ϕ) does not contain yr, rs or st since η ∈ ¯ϕ(t).
+Let ϕ1 = ϕ/Px(1, η, ϕ) and we are back to the case when η = 1.
+If β = 1, then Px(η, 1, ϕ) and Pt(η, 1, ϕ) are disjoint. If Px(η, 1, ϕ) does not pass through
+rs, let ϕ1 = ϕ/Px(η, 1, ϕ). Then η is missing at y1 now and we are back to the case when
+η = 1. If Pt(η, 1, ϕ) does not contain rs, let ϕ1 = ϕ/Pt(η, 1, ϕ). Then β ∈ ¯ϕ1(t) and we are
+back to Case B.1. This completes the proof of Case B.2, and so Claim B holds.
+By Claims A and B, we assume that ϕ satisfies the following properties:
+• ϕ(yr) = 3 ∈ ¯ϕ(t), ϕ(rs) = β ∈ ¯ϕ(t).
+• ϕ(st) = γ ̸= 3
+Claim C. We may further assume β = ϕ(yz) = 2.
+Proof. Suppose to the contrary β ̸= 2.
+Case C.1: γ ̸= ϕ(yz) (i.e. γ ̸= 2).
+Case C.1.1: 1 ∈ {γ, β}.
+If β = 1, then Pt(γ, 1, ϕ) does not pass through x or y. Let ϕ1 = ϕ/Pt(γ, 1, ϕ). Then
+ϕ1(st) = 1. Thus we assume γ = 1.
+If β ∈ ¯ϕ(z), let ϕ1 = ϕ/Px(β, 1, ϕ) and then recolor yz with β. Then ϕ1 is a desired
+coloring.
+If 3 ∈ ¯ϕ(z), let ϕ1 = ϕ/Px(β, 1, ϕ) and ϕ2 = ϕ1/Pz(3, β, ϕ1). Notice that the second
+Kempe exchange will not effect yr or rs since they are on Px(3, β, ϕ1) = Py(3, β, ϕ1) by
+Lemma 5.1. Thus we obtain a desired coloring by recoloring yz with β under ϕ2.
+Now we assume 3, β ̸∈ ¯ϕ(z).
+If ¯ϕ(z) ∩ ¯ϕ(t) ̸= ∅, let η ∈ ¯ϕ(z) ∩ ¯ϕ(t).
+Then η ̸∈ {1, 2, 3, β} and η ∈ ¯ϕ(x).
+Note
+that Px(1, η, ϕ) = Py(1, η, ϕ) does not contain st since η ∈ ¯ϕ(t). Let ϕ1 = ϕ/Px(1, η, ϕ) and
+then η ∈ ¯ϕ1(y). Let ϕ2 = ϕ1/Pz(η, 3, ϕ1) and then 3 ∈ ¯ϕ2(z). Note that Pz(η, 3, ϕ1) does
+not contain yr or t since yr is on Px(η, 3, ϕ1) = Py(η, 3, ϕ1) and 3, η ∈ ¯ϕ1(t). Finally let
+ϕ3 = ϕ2/Px(η, 1, ϕ2). We are back to the case when ϕ(yr) ∈ ¯ϕ(z).
+16
+
+Now assume ¯ϕ(z)∩ ¯ϕ(t) = ∅. Since d(z)+d(t) ≤ ∆, ϕ(z) and ϕ(t) form a partition of C.
+Consequently, we have 1 ∈ ¯ϕ(z) and 2 ∈ ¯ϕ(t). Since d(z) ≤ ∆ − 2, let η ∈ ¯ϕ(z) \ {1}. Clearly
+η ∈ ¯ϕ(x) and η /∈ {1, 2, 3, β}. Let ϕ1 be the coloring obtained from ϕ by recoloring yz with 1.
+Then 2 ∈ ¯ϕ1(y)∩ ¯ϕ1(z) and Px(2, η, ϕ1) = Py(2, η, ϕ1) by Lemma 5.1. Let ϕ2 = ϕ1/Px(2, η, ϕ1)
+and ϕ3 be the coloring obtained from ϕ2 by recoloring yz with η. Now we have γ = 1 ∈ ¯ϕ3(y),
+ϕ3(yz) = η ̸= β and 2 ∈ ¯ϕ3(z) ∩ ¯ϕ3(t). Thus we are back to the case when ¯ϕ(z) ∩ ¯ϕ(t) ̸= ∅.
+This completes the proof of Case C.1.1.
+Case C.1.2: 1 /∈ {γ, β}.
+Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t)\{2, 3, β}. We may assume η = 1. Otherwise, η ∈ ¯ϕ(x)
+since ¯ϕ(x) = C\{1, 2}. Thus by interchanging colors on Pt(1, η, ϕ), 1 is missing at t. Since
+γ ∈ ¯ϕ(x), we have Px(γ, 1, ϕ) = Py(γ, 1, ϕ). Since 1 ∈ ¯ϕ(t), Px(γ, 1, ϕ) does not contain st.
+Therefore, by interchanging γ and 1 on Px(γ, 1, ϕ), we are back to Case C.1.1. This completes
+the proof of Case C.1.
+Case C.2: γ = ϕ(yz) = 2.
+In this case, ϕ(yz) = ϕ(st) = 2 ∈ ϕ(z) ∩ ϕ(t). If 1 ∈ ¯ϕ(z), recolor yz with 1. Then we are
+back to Case C.1 if β ̸= 1. Otherwise, we have a desired coloring. Thus in the following we
+assume 1 ∈ ϕ(z).
+Case C.2.1: {3, β} ∩ ¯ϕ(z) ̸= ∅.
+If β ∈ ¯ϕ(z), then by Lemma 5.2, Pz(2, β, ϕ) ends at x since β ∈ ¯ϕ(x) and it is disjoint
+from Pt(2, β, ϕ). Thus ϕ1 = ϕ/Pz(2, β, ϕ) is a desired coloring.
+Assume 3 ∈ ¯ϕ(z) and β ∈ ϕ(z).
+If β = 1, then Py(1, 3, ϕ) contains the edges yr and rs and is disjoint from Pz(1, 3, ϕ).
+Note that 1, β ∈ ¯ϕ(t). Let ϕ1 = ϕ/Pz(1, 3, ϕ) and we are back to the case when 1 ∈ ¯ϕ(z).
+Assume β ̸= 1. Since d(z) ≤ ∆ − 2, let η ∈ ¯ϕ(z) \ {3}. Then η ̸∈ {1, 2, 3, β}. Thus
+Pz(1, η, ϕ) does not contain the vertices x, y or the edges rs, st. Let ϕ1 = ϕ/Pz(1, η, ϕ) and
+we are back to the case when 1 ∈ ¯ϕ(z). This completes the proof of Case C.2.1.
+Case C.2.2: {3, β} ∩ ¯ϕ(z) = ∅.
+Since 2 ∈ ϕ(z) ∩ ϕ(t) and d(z) + d(t) ≤ ∆, let η ∈ ¯ϕ(t) ∩ ¯ϕ(z). Then η ∈ ¯ϕ(x). If
+β ̸= 1, by interchanging colors on Px(η, 1, ϕ) and then recoloring yz with η, we are back to
+Case C.1. Suppose β = 1. Then Px(η, 1, ϕ) and Pz(η, 1, ϕ) are disjoint and either Px(η, 1, ϕ)
+or Pz(η, 1, ϕ) does not contain rs. In the former case, by interchanging η and 1 on Px(η, 1, ϕ)
+and then recoloring yz with η, we are back to Case C.1. In the later case by interchanging η
+and 1 on Pz(η, 1, ϕ) and then recoloring yz with 1, we have a desired coloring. This completes
+the proof of Case C.2.2, and so Claim C holds.
+By Claim C, we further assume ϕ(yz) = ϕ(rs) = 2. Note that ϕ(x) ∩ ϕ(y) = {2} and
+¯ϕ(x) ∪ ¯ϕ(y) = C\{2}.
+Claim D. We may further assume that ¯ϕ(y) ∩ ¯ϕ(z) ̸= ∅ and γ ∈ ¯ϕ(y) ∩ ¯ϕ(z).
+That is
+γ = 1 ∈ ¯ϕ(z).
+Proof. We split the proof into the following cases.
+17
+
+Case D.1: ϕ(yr) = 3 ∈ ¯ϕ(z).
+Case D.1.1: γ = 1.
+In this case Px(1, 3, ϕ) is disjoint from Pz(1, 3, ϕ). Let ϕ1 = ϕ/Pz(1, 3, ϕ). If Pz(1, 3, ϕ)
+does not end at t, then ϕ1 is a desired coloring. If Pz(1, 3, ϕ) ends at t, let ϕ2 be the coloring
+obtained from ϕ1 by recoloring yz with 1. In the coloring ϕ2, 2 is missing at y, 3 is missing
+at x, and Py(3, 2, ϕ2) = yrst, a contradiction to Lemma 5.1. This proves Case D.1.1.
+Case D.1.2: γ ̸= 1. Then γ ̸∈ {1, 2, 3} and γ ∈ ¯ϕ(x).
+If 1 ∈ ¯ϕ(t), then Px(1, γ, ϕ) ends at y and thus does not contain the edge st. Thus by
+interchanging 1 and γ on Px(1, γ, ϕ), we are back to Case D.1.1.
+Assume 1 ̸∈ ¯ϕ(t). Since d(t) ≤ ∆ − 4, let η ∈ ¯ϕ(t)\{2, 3}. Then η ̸∈ {1, 2, 3, γ} and
+η ∈ ¯ϕ(x). By interchanging the colors on Pt(η, 1, ϕ), we are back to the case when 1 ∈ ¯ϕ(t).
+This proves Case D.1.
+Case D.2: ϕ(yr) = 3 /∈ ¯ϕ(z).
+Since d(z) + d(t) ≤ ∆, either ϕ(z) and ϕ(t) form a partition of C or there exists a color
+η ∈ ¯ϕ(z) ∩ ¯ϕ(t).
+Case D.2.1: There exists a color η ∈ ¯ϕ(z) ∩ ¯ϕ(t).
+In this case we have η /∈ {2, 3, γ} and η ∈ ¯ϕ(x) ∪ ¯ϕ(y).
+If η = 1, then Pz(1, 3, ϕ) does not pass through x, y or t since both α and η are missing
+at t. We are back to Case D.1 by interchanging 1 and 3 on Pz(1, 3, ϕ).
+If η ̸= 1, then η ∈ ¯ϕ(x) and Px(η, 1, ϕ) does not pass through t since η ∈ ¯ϕ(t) ∩ ¯ϕ(z).
+Thus by interchanging η and 1 on Px(η, 1, ϕ), we are back to the case when η = 1. This
+completes the proof of Case D.2.1.
+Case D.2.2: ϕ(z) and ϕ(t) form a partition of C.
+In this case γ ∈ ¯ϕ(z). If γ = 1, then ϕ is a desired coloring. Therefore we assume
+γ ̸= 1.
+Thus γ ∈ ¯ϕ(x).
+Let η ∈ ¯ϕ(t)\{2, 3}.
+By Lemma 5.1, Px(1, η, ϕ) does not pass
+through z or t. Note that if 1 = η, then Px(1, η, ϕ) = x. Let ϕ1 = ϕ/Px(1, η, ϕ). Then
+Px(η, γ, ϕ1) = Py(η, γ, ϕ1). Note that Px(η, γ, ϕ1) does not contain t since η ∈ ¯ϕ1(t). Let
+ϕ2 = ϕ1/Px(η, γ, ϕ1). Then we have γ ∈ ¯ϕ2(y) ∩ ¯ϕ2(z) and thus ϕ1 is a desired coloring. This
+completes the proof of Case D.2, and so Claim D holds.
+In summary, by Claims A, B, C, and D, we assume that ϕ satisfies the following properties:
+• ϕ(x) = {1, 2} and 1 ∈ ¯ϕ(y) ∩ ¯ϕ(z)
+• ϕ(yr) = 3, ϕ(yz) = ϕ(rs) = 2, and ϕ(st) = 1
+• 2, 3 ∈ ¯ϕ(t).
+Note that Px(1, 3, ϕ) ends at y and is disjoint from Pt(1, 3, ϕ).
+If Pt(1, 3, ϕ) does not
+end at z, let ϕ1 be the coloring obtained from ϕ by interchanging colors on Pt(1, 3, ϕ) and
+recoloring yz with 1. Then 3 ∈ ¯ϕ1(x), 2 ∈ ¯ϕ1(y) and Py(3, 2, ϕ1) = yrst not ending at x,
+a contradiction to Lemma 5.1. Thus Pt(1, 3, ϕ) ends at z. Let ϕ2 = ϕ/Pt(1, 3, ϕ). Then
+Pz(2, 3, ϕ2) = zyrst which does not end at x, a contradiction to Lemma 5.2. This completes
+the proof of Lemma 2.9.
+18
+
+5.4
+Proof of Lemma 2.11
+Lemma 2.11 Let G be a ∆-critical graph and xy be an edge with d(x) + d(y) = ∆ + 3 and
+max{d(x), d(y)} < ∆. Then x has d(x)−2 neighbors of degree ∆ having no (∆−2)−-neighbors
+other than x, y.
+Proof.
+Let ϕ ∈ C∆(G − xy).
+Since G is ∆-critical and d(x) + d(y) = ∆ + 3, we have
+|ϕ(x) ∩ ϕ(y)| = 1.
+Let δ be the color in ϕ(x) ∩ ϕ(y).
+Then ¯ϕ(x) ∪ ¯ϕ(y) = C\{δ}.
+By
+Lemma 2.1, x has at least d(x) − 2 neighbors of degree ∆. Thus including y, x has at most
+two neighbors of degree less than ∆. By Lemmas 5.3 and 5.2, we have the following fact which
+will be applied frequently.
+Fact 1. Let yxzt be a path with ϕ(xz) ∈ ¯ϕ(y).
+(1) ¯ϕ(z) ⊆ {δ} and thus d(z) ≥ ∆ − 1. If δ ∈ ¯ϕ(z), then for any color η ∈ ϕ(z) \ {ϕ(xz)},
+Pz(δ, η, ϕ) ends at x or y.
+(2) If yxzt is a Kierstead path, then ¯ϕ(t) ⊆ {δ} and thus d(t) ≥ ∆ − 1.
+We consider two cases in the following according to the number of ∆-neighbors of x.
+Case 1. x has a neighbor z0 ̸= y with d(z0) < ∆.
+It is sufficient to show that for any path yxzt with z ̸= z0, we have d(t) ≥ ∆ − 1.
+Suppose to the contrary that there is a path yxzt such that z ̸= z0 but d(t) ≤ ∆ − 2. We
+consider two cases according to ϕ(xz0) = δ or not.
+Case 1.1: α = ϕ(xz0) ̸= δ.
+By Fact 1(1), ¯ϕ(z0) = {δ}.
+First assume ϕ(xz) ∈ ¯ϕ(y). Then by Fact 1(2), ϕ(zt) = δ otherwise yxzt is a Kierstead
+path. Since d(t) ≤ ∆ − 2 and δ ∈ ϕ(t), let η ∈ ¯ϕ(t) \ {α}. By Fact 1(1), Pz0(δ, η, ϕ) ends at
+x or y and thus is disjoint from Pt(δ, η, ϕ). Let ϕ1 = ϕ/Pt(δ, η, ϕ). Then yxzt is a Kierstead
+path in ϕ1 and thus d(t) ≥ ∆ − 1 by Fact 1(2), a contradiction.
+Now assume ϕ(xz) = δ. Denote β = ϕ(zt). Then β ∈ ¯ϕ(x) ∪ ¯ϕ(y). We may assume
+that β ∈ ¯ϕ(x). Otherwise if there is a color η ∈ ¯ϕ(t) ∩ ¯ϕ(x), interchange colors on the path
+Pt(η, β, ϕ) which does not contain x or y. If no such η exists, let η ∈ ¯ϕ(x) and γ ∈ ¯ϕ(t) \ {δ}.
+Let ϕ1 = ϕ/Pt(η, γ, ϕ) and then let ϕ2 = ϕ1/Pt(η, β, ϕ1).
+By Fact 1(1), Pz0(δ, β, ϕ) ends at x and thus contains xzt. This implies δ ∈ ϕ(t). Thus
+| ¯ϕ(t) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y))| ≥ 2 since d(t) ≤ ∆ − 2. Let η ∈ ¯ϕ(t) \ {α}. By Fact 1(1) again,
+Pz0(δ, η, ϕ) ends at x or y and thus is disjoint from Pt(δ, η, ϕ). Let ϕ1 = ϕ/Pt(δ, η, ϕ). Then
+in ϕ1, Px(δ, β, ϕ1) = xzt which is disjoint from Pz0(δ, β, ϕ1), a contradiction to Fact 1(1). This
+completes the proof of Case 1.1.
+Case 1.2: ϕ(xz0) = δ.
+Then ϕ(xz) ∈ ¯ϕ(y). Since d(t) ≤ ∆−2, by Fact 1(2), ϕ(zt) = δ. Let η ∈ ¯ϕ(z0). Similar to
+the argument in Case 1.1, we assume η ∈ ¯ϕ(x). Recolor xz0 with η. Then yxzt is a Kierstead
+path. By Fact 1(2), d(t) ≥ ∆ − 1, a contradiction. This completes the proof of Case 1.
+Case 2. All vertices in N(x)\{y} are ∆-vertices.
+19
+
+Since | ¯ϕ(y)∩ϕ(x)| = d(x)−2, we are done if d(t) ≥ ∆−1 for every path yxzt with ϕ(xz) ∈
+¯ϕ(y). Thus assume that there is a path yxz0t0 such that ϕ(xz0) ∈ ¯ϕ(y) and d(t) ≤ ∆ − 2.
+By Fact 1(2), ϕ(z0t0) = δ. Denote α = ϕ(xz0). Then α ∈ ¯ϕ(y). With a similar argument as
+before, we may assume α ∈ ¯ϕ(t0) and there is a color η ∈ ¯ϕ(t0) ∩ ¯ϕ(x). Then η ̸= α. Now it
+is sufficient to show that for any path yxzt with z ̸= z0, we have d(t) ≥ ∆ − 1. We consider
+the following two cases.
+Case 2.1. ϕ(xz) = β ∈ ¯ϕ(y).
+Then by Fact 1(2), ϕ(zt) = δ, so t ̸= t0. Since d(t) ≤ ∆ − 2, there is a color η1 ∈ ¯ϕ(t).
+Then η1 ∈ ¯ϕ(x) ∪ ¯ϕ(y). Similarly we may assume η, η1 ∈ ¯ϕ(x). Note that d(t) ≤ ∆ − 2 and
+d(t0) ≤ ∆ − 2. Thus η ̸= η1 since otherwise both Pt0(δ, η, ϕ) and Pt(δ, η, ϕ) end at x by Fact
+1(2), a contradiction.
+Now let ϕ1 be the coloring obtained from ϕ by coloring xy with α, leaving xz0 uncolored
+and recoloring z0t0 with α.
+Then Px(η1, δ, ϕ1) = Pz0(η1, δ, ϕ1) by Lemma 5.1.
+Let ϕ2 =
+ϕ1/Pt(η1, δ, ϕ1). Then ϕ2(zt) = η1 ∈ ¯ϕ2(x). Note that the last Kempe exchange may affect
+the colors of the edges incident to t0, so δ may not be missing at t0 under ϕ2. But we still
+have η ∈ ¯ϕ2(x) ∩ ¯ϕ2(t0). If δ ∈ ϕ2(t0), let ϕ3 = ϕ2/Pt0(η, δ, ϕ2). Otherwise let ϕ3 = ϕ2. Then
+we have δ ∈ ¯ϕ3(z0)∩ ¯ϕ3(t0). Finally let ϕ4 be the coloring obtained from ϕ3 by recoloring z0t0
+with δ, coloring xz0 with α and leaving xy uncolored. Then yxzt is a Kierstead path under
+ϕ4. However d(t) ≤ ∆ − 2, a contradiction to Fact 1(2).
+Case 2.2 ϕ(xz) = δ.
+Denote ϕ(zt) = β. With similar arguments as before we may assume that there is a color
+η′ ∈ ¯ϕ(t) ∩ ¯ϕ(x). We may then assume that β ∈ ¯ϕ(x) since otherwise we can interchange β
+and η′ on Px(β, η′, ϕ) to get a desired coloring.
+Since d(t) ≤ ∆−2, let η1 ∈ ¯ϕ(t)\{α}. We then show that we may assume η1 ∈ ¯ϕ(x)∪{δ}.
+Suppose otherwise η1 ∈ ¯ϕ(y)\{α}. Since d(x) ≤ ∆ − 1, we have | ¯ϕ(x)| ≥ 2. Let α′ be a color
+in ¯ϕ(x)\{ϕ(zt)}. By interchanging η1 and α′ on Px(η1, α′, ϕ), we obtain a coloring as desired.
+Let ϕ1 be the coloring obtained from ϕ by coloring xy with α, leaving xz0 uncolored and
+recoloring z0t0 with α. Then under ϕ1, z0xzt is a Kierstead path with η1 ∈ ( ¯ϕ1(x) ∪ ¯ϕ1(z0)) ∩
+¯ϕ1(t), a contradiction to Lemma 5.3. This completes the proof of the lemma.
+5.5
+Proof of Lemma 2.12
+Lemma 2.12 Let G be a 7-critical graph and x be a 5-vertex.
+(1) if x has three 6-neighbors, then each 7-neighbor of x has exactly one 5−-neighbor.
+(2) if x has two 6-neighbors, then x has two 7-neighbors, each of which has at most two
+5−-neighbors.
+(3) if x has exactly four 7-neighbors, then x has two 7-neighbors, each of which has at most
+three 5−-neighbors.
+Proof. If x has a 5-neighbor, then by Lemma 2.1, x has at least three 7-neighbors and thus
+has at most one 6-neighbor. To show the lemma in this case, we only need to consider the case
+20
+
+when x has four 7-neighbors and one 5-neighbor which is (3), and it follows from Lemma 2.11.
+In the rest of the proof, we assume that x has no 5-neighbors. By the assumption of the
+lemma, x has a 6-neighbor. Let y be a 6-neighbor of x, ϕ ∈ C∆(G − xy). Without loss of
+generality we assume that ¯ϕ(y) = {1, 2}, ¯ϕ(x) = {3, 4, 5}, and ϕ(x) ∩ ϕ(y) = {6, 7}. By
+Lemma 2.1, x has at least two 7-neighbors.
+(1) Denote the two 6-vertices in N(x)\{y} by z1, z2, the two 7-vertices in N(x) by v1, v2. We
+need to show that for any path yxvt with v ∈ {v1, v2}, d(t) ≤ 5. We consider three cases.
+Case 1.1 x, y, z1, z2 form the vertex set of a multi-fan with respect to xy and ϕ.
+In this case, by Lemma 5.1, we have ¯ϕ(z1) ∪ ¯ϕ(z2) = {6, 7}.
+Assume without loss of
+generality that ¯ϕ(z1) = {6} and ¯ϕ(z2) = {7}. Then for each α ∈ ¯ϕ(x)∪ ¯ϕ(y), both Pz1(6, α, ϕ)
+and Pz2(7, α, ϕ) end at x if α ∈ ¯ϕ(x).
+Let yxvt be a path where d(v) = 7. Let η be a color in ¯ϕ(t) and β = ϕ(vt). We may
+assume that η ∈ ¯ϕ(x) since otherwise η ∈ {1, 2, 6, 7}, and we can interchange η and 3 on
+Pt(η, 3, ϕ), which doesn’t pass through x or y by Lemma 5.1, to obtain a desired coloring.
+Thus we assume η ∈ ¯ϕ(x).
+We may further assume that β = ϕ(v1t) ∈ ¯ϕ(x). Otherwise β ∈ {1, 2, 6, 7}. Note that
+Pt(β, η, ϕ) does not end at x or y. Let α ∈ ¯ϕ(x) \ {η}. Interchange η and ϕ(vt) = β on
+Pt(β, η, ϕ) first and then interchange β, α on the (β, α)-chain starting at t.
+We obtain a
+desired coloring. Thus we assume that β ∈ ¯ϕ(x).
+Now let ϕ1 = ϕ/Pt(η, ϕ(xv), ϕ). Then ϕ(xv) ∈ ¯ϕ1(t) and Px(ϕ(xv), ϕ(vt), ϕ1) = xvt does
+not end at y, z1, or z2, a contradiction to Lemma 5.1. This completes the proof of Case 1.1.
+Case 1.2 x, y, z1, z2 do not form the vertex set of a multi-fan with respect to xy and ϕ,
+and |{ϕ(xz1), ϕ(xz2)} ∩ {1, 2}| = 1.
+By symmetry, assume that ϕ(xz1) = 1, ¯ϕ(z1) = {6}, ϕ(xz2) = 7, ϕ(xv1) = 2, and
+ϕ(xv2) = 6. Then for each color η ∈ {2, 3, 4, 5}, Pz1(η, 6, ϕ) ends at x or y depending on
+whether η ∈ ¯ϕ(x) or η ∈ ¯ϕ(y) by Lemma 5.1. Similar to the argument in Case 1.1, we may
+further assume 3 ∈ ¯ϕ(z2).
+Let yxvt be a path where d(v) = 7. Then ϕ(xv) ∈ {2, 6}. We first assume ϕ(xv) = 2.
+If ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y), then yxvt is a Kierstead path with d(x) < ∆. Thus ¯ϕ(t1) = {6, 7}
+by Lemma 5.3. Let η be a color in ¯ϕ(x) \ {ϕ(vt)}. Then by Lemma 5.4, Pt(η, 6, ϕ) ends at x.
+However, by Lemma 5.1, Pz1(η, 6, ϕ) ends at x, a contradiction.
+If ϕ(vt) = 7, recolor xz2 with 3 and we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y).
+If ϕ(vt) = 6, let η ∈ ¯ϕ(t) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)). We may assume η ∈ ¯ϕ(x) since otherwise we
+can pick a color β ∈ ¯ϕ(x) and interchange colors on Pt(η, β, ϕ). Since Pz1(η, 6, ϕ) ends at x,
+Pt(η, 6, ϕ) and Pz1(η, 6, ϕ) are disjoint. Interchange colors on Pt(η, 6, ϕ) and we are back to
+the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y) again.
+Now we assume ϕ(xv) = 6. Denote ϕ(vt) = β. If β = 7, then recolor the edge xz2 with 3
+and then 7 is missing at x. Thus we may assume β ∈ ¯ϕ(x) ∪ ¯ϕ(y).
+If ϕ(vt) = β ∈ ¯ϕ(x), then Px(6, β, ϕ) ends at z1 and thus 6 ∈ ϕ(t). Since d(t) ≤ 5, let
+21
+
+α ∈ ¯ϕ(t) ∩ ( ¯ϕ(x) ∪ ¯ϕ(y)). Similarly as before we may further assume that α ∈ ¯ϕ(x). Note
+that Pz1(α, 6, ϕ) and Pt(α, 6, ϕ) are disjoint. Let ϕ1 = ϕ/Pt(α, 6, ϕ). Then 6 is missing at t
+and thus Px(6, β, ϕ1) = xvt does not end at z1, a contradiction.
+Suppose ϕ(vt) = β ∈ ¯ϕ(y). Let α′ be a color in ¯ϕ(t)\{7}. Then similarly, we can assume
+that α′ ∈ ¯ϕ(x). By interchanging α′ and β on Pt(α′, β, ϕ), we are back to the case when
+ϕ(vt) ∈ ¯ϕ(x). This completes the proof of Case 1.2.
+Case 1.3 {ϕ(xz1), ϕ(xz2)} = {6, 7}.
+Let yxvt be a path where d(v) = 7.
+Without loss of generality, assume ϕ(xz1) = 6,
+ϕ(xz2) = 7, and ϕ(xv) = 1. Denote ϕ(vt) = β.
+We first assume β ∈ ¯ϕ(x) ∪ ¯ϕ(y). Then yxvt is a Kierstead path with d(x) < ∆. Thus
+¯ϕ(t) = {6, 7} by Lemma 5.3. Let α be a color in ¯ϕ(x)\{β} and η be a color in ¯ϕ(z1). Note
+that Px(α, 7, ϕ) ends at t by Lemma 5.4. Thus we may assume that η ∈ ¯ϕ(x) since otherwise
+η ∈ {1, 2, 7} and we can interchange η, α on Pz1(η, α, ϕ). So we assume η ∈ ¯ϕ(x). We then
+claim that we may further assume that η ∈ ¯ϕ(x)\{ϕ(vt)}. Otherwise η = ϕ(vt) ∈ ¯ϕ(x).
+Interchange η, 1 on Pz1(η, 1, ϕ) first and then interchange 1, α on the (1, α)-chain starting
+at z1.
+Thus we assume that η ∈ ¯ϕ(x)\{ϕ(vt)}.
+Now Px(η, 6, ϕ) ends at z1 but not t, a
+contradiction to Lemma 5.4.
+Now we further assume β ∈ {6, 7}. Without loss of generality assume ϕ(vt) = 6. Let
+η′ ∈ ¯ϕ(t)\{7}. With a similar argument as before, we assume η′ ∈ ¯ϕ(x). Let η1 be the color
+missing at z1 and η2 be the color missing at z2.
+We first claim η1 = 7. Since otherwise, we have η1 ∈ {1, 2, 3, 4, 5} and by interchanging
+η1, 3 on Pz1(η1, 3, ϕ) if necessary, we may assume that η1 ∈ ¯ϕ(x). Then by recoloring xz1 with
+η1, we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y).
+We then claim η2 = 6.
+Since otherwise, η2 ∈ {1, 2, 3, 4, 5} and by interchanging η2, 3
+on Pz2(η2, 3, ϕ) if necessary, we may assume that η2 ∈ ¯ϕ(x).
+By recoloring xz2 with η2
+and then recoloring xz1 with 7, we are back to the case when ϕ(vt) ∈ ¯ϕ(x) ∪ ¯ϕ(y). Thus
+η2 = 6. Note that the above argument also implies that Pz2(6, η′, ϕ) ends at x, since otherwise
+by interchanging 6, η′ on this path, we are back to the case when η2 ̸= 6. Now let ϕ1 =
+ϕ/Pt(η′, 6, ϕ), we have ϕ1(vt) = η′ ∈ ¯ϕ1(x), and thus we are back to the case when ϕ(vt) ∈
+¯ϕ(x) ∪ ¯ϕ(y). This completes the proof of (1).
+□
+(2) Since x has no 5−-neighbors, by (1) x has two 6-neighbors and three 7-neighbors. Denote
+by v1, v2, v3 the three 7-vertices and z the 6-neighbor of x distinct from y. Then ϕ(xz) ∈ {1, 2}
+or ϕ(xz) ∈ {6, 7}.
+Case 2.1 ϕ(xz) ∈ {1, 2}.
+In this case, x, y, z form the vertex set of a multi-fan with respect to xy and ϕ.
+By
+Lemma 5.1, we have ¯ϕ(z) ∈ {6, 7}.
+Assume without loss of generality that ϕ(xz) = 1,
+¯ϕ(z) = {6}, ϕ(xv1) = 2 and ϕ(xv2) = 6. Note that if each of v1 and v2 has at most two
+5−-neighbors, then we are done. Thus we consider the following two cases.
+If v1 has three 5−-neighbors, then there exists t1 ∈ N(v1)\{x} such that d(t1) ≤ 5 and
+22
+
+ϕ(v1t1) ̸= 7. Let η1 be a color in ¯ϕ(t1)\{7}. With similar arguments as before we may assume
+that η1 ∈ ¯ϕ(x) and ϕ(v1t1) ∈ ¯ϕ(x). Now yxv1t1 is a Kierstead path with respect to xy and
+ϕ. But η1 ∈ ¯ϕ(x) ∩ ¯ϕ(t1), a contradiction to Lemma 5.3.
+If v2 has three 5−-neighbors, then there exists t2 ∈ N(v2)\{x} such that d(t1) ≤ 5 and
+ϕ(v2t2) ̸= 7. Let η2 be a color in ¯ϕ(t2)\{7}. Similar to the argument before, we may assume
+that η2 and ϕ(v2t2) are in ¯ϕ(x). Let ϕ′ = ϕ/Pt2(η2, 6, ϕ). Then we have 6 ∈ ¯ϕ′(t2). Thus
+Px(6, ϕ′(v2t2), ϕ′) = xv2t2 does not end at z, a contradiction to Lemma 5.1. This completes
+the proof of Case 2.1.
+Case 2.2 ϕ(xz) ∈ {6, 7}.
+In this case, we may assume without loss of generality that ϕ(xz) = 6, ϕ(xv1) = 1 and
+ϕ(xv2) = 2. Note that if each of v1 and v2 has at most two 5−-neighbors, then we are done.
+Thus by the symmetry, assume that v1 has three 5−-neighbors. Then there exist two vertices
+t, t′ ∈ N(v1)\{x} such that d(t) ≤ 5 and d(t′) ≤ 5.
+Claim 1 {ϕ(v1t), ϕ(v1t′)} = {6, 7}.
+Otherwise, without loss of generality, assume ϕ(v1t) ∈ ¯ϕ(x) ∪ ¯ϕ(y). Then y, x, v1, t form
+the vertex set of a Kierstead path with d(x) < ∆. Thus ¯ϕ(t) = {6, 7} by Lemma 5.3. Let
+α be a color in ¯ϕ(x)\{ϕ(v1t)} and η be the color in ¯ϕ(z). Note that Px(α, 7, ϕ) ends at t
+by Lemma 5.4. Thus we may assume that η ∈ ¯ϕ(x) since otherwise η ∈ {1, 2, 7} and we
+can interchange η, α on Pz1(η, α, ϕ). Furthermore, we may assume that η ∈ ¯ϕ(x)\{ϕ(v1t)}.
+Otherwise η = ϕ(v1t) ∈ ¯ϕ(x), and we can interchange η, 1 on Pz(η, 1, ϕ) first and then
+interchange 1, α on the (1, α)-chain starting at z. Now the (6, η)-chain starting at x ends at
+z but not t, a contradiction to Lemma 5.4. Therefore {ϕ(v1t), ϕ(v1t′)} = {6, 7} and without
+loss of generality, we assume that ϕ(v1t) = 6 and ϕ(v1t′) = 7. This completes the proof of
+Claim 1.
+Claim 2 ¯ϕ1(z) ̸= {7}.
+Let η be the color missing at z. . Otherwise η ∈ ¯ϕ(x) ∪ ¯ϕ(y). We may assume that
+η ∈ ¯ϕ(x) since otherwise we can interchange η and 3 on Pz(η, 3, ϕ) to get the desired coloring.
+Now by recoloring xz with η, we have {ϕ(v1t), ϕ(v1t′)} ̸= {6, 7}, a contradiction to Claim 1.
+Thus ¯ϕ(z) = {7}.
+Now let η′ be a color in ¯ϕ(t′)\{6}. Similarly as before, we may assume that η′ ∈ ¯ϕ(x). If
+Pt′(η′, 7, ϕ) does not end at x, let ϕ1 = ϕ/Pt′(η′, 7, ϕ). Then we have {ϕ1(v1t), ϕ2(v1t′)} ̸=
+{6, 7}, a contradiction to Claim 1. If Pt′(η′, 7, ϕ) ends at x, let ϕ1 = ϕ/Pz(η′, 7, ϕ). Then we
+have ¯ϕ1(z) ̸= {7}, a contradiction to Claim 2. This completes the proof of (2).
+□
+(3) Since y is the only 6-neighbor of x and |ϕ(x) ∩ ϕ(y)| = 2, there are two 7-neighbors of x,
+say v1, v2, such that {ϕ(xv1), ϕ(xv2)} ⊆ ¯ϕ(y). It is sufficient to show that each v1 and v2 has
+at most three 5−-neighbors.
+Suppose to the contrary that v1 has three 5���-neighbors other than x, say t1, t2, t3. Since
+| ¯ϕ(x)| = 3, | ¯ϕ(y)| ≥ 2 and | ¯ϕ(ti)| ≥ 2 for each i = 1, 2, 3, by Lemma 5.5, at most one
+of ϕ(v1t1), ϕ(v1t2), ϕ(v1t3) is in ¯ϕ(x) ∪ ¯ϕ(y). Without loss of generality, assume ϕ(v1t1) ∈
+23
+
+¯ϕ(x)∪ ¯ϕ(y). Then {ϕ(v1t2), ϕ(v1t3)} = {6, 7}. By Lemma 5.3, we have ¯ϕ(t1) = ϕ(x)∩ϕ(y) =
+{6, 7}. Thus {y, x, v1, t1, t2, t3} is the vertex set of a ϕ-broom. But {y, x, v1, t1, t2, t3} is not
+elementary, a contradiction to Lemma 5.5. This completes the proof of (3) and thus completes
+the proof of the lemma.
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+25
+

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+Dark matter freeze-in via a light thermal fermion
+mediator
+Shao-Ping Lia
+aInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
+E-mail: [email protected]
+Abstract: The connection between a hidden nonthermal sector and a thermal plasma
+can be established by a light fermion mediator, which was once thermalized in the early
+universe. When the mediator is much lighter than the lowest scale in the hidden sector,
+both the kinematically forbidden decay and the scattering can coexist to produce the
+hidden species at the same order of coupling constants. This work serves to present a
+dedicated investigation into the freeze-in dark matter production via a thermalized fermion
+mediator, taking into account consistently the forbidden decay and scattering channels. We
+demonstrate that the plasma-induced decay rate generically differs from that calculated
+via the tree-level amplitude, but the former can be simply estimated from the latter with
+constant rescaling. While the contribution to the dark matter relic density is dominated by
+the scattering channel, the portion from the forbidden decay can reach 40% in the weak-
+coupling regime and hence cannot be ignored for a precise prediction of the relic density.
+This work also provides a simple method to estimate the relative effect of the scattering
+and the forbidden decay.
+arXiv:2301.02835v1  [hep-ph]  7 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+The simplified scenario
+3
+3
+Forbidden decay
+4
+3.1
+Boltzmann equation
+4
+3.2
+Spectral density of the fermion mediator
+5
+3.3
+Collision rate
+7
+3.3.1
+One-loop retarded amplitude
+7
+3.3.2
+Tree-level amplitude
+9
+4
+Scattering
+11
+4.1
+Double counting and resonant enhancement
+11
+4.2
+Tree-level scattering amplitude without thermal correction
+12
+5
+DM relic density
+14
+6
+Realistic scenarios and possible signals
+16
+7
+Conclusions
+18
+A Thermal one-loop amplitudes
+19
+A.1 The DM part
+19
+A.2 The fermion mediator part
+19
+1
+Introduction
+A hidden nonthermal species can be created in the early universe from the thermal plasma
+via a light mediator [1]. If the hidden sector consists of feebly interacting dark matter (DM),
+the direct DM detection could be very challenging. However, a light mediator connecting
+the DM with the standard model (SM) can provide an indirect avenue to test the feeble
+DM scenarios if the connection between the mediator and the SM is relatively strong.
+The phenomenology of DM production from a light mediator is fruitful, e.g., the mil-
+licharged DM production from a vector mediator [2–5] and the sterile neutrino DM pro-
+duction via a scalar mediator [6–10]. There are also many interesting DM scenarios via a
+fermion mediator. A typical example is that the sterile neutrino itself can be the mediator
+to connect a stable dark sector with the SM particles [11–21].
+In general, the DM production in the early universe depends on the relative mass of
+the mediator and the dark sector. For a light mediator, however, when the mediator has
+– 1 –
+
+a vacuum mass larger than the dark sector, the mediator decay plays the dominant role
+in generating the DM relic density, while the scattering effect is usually subdominant or
+negligible due to the suppression of higher-order weak couplings and additional phase-space
+factors. If the mediator is much lighter than the dark sector, the decay channel is kine-
+matically forbidden in vacuum and the scattering/annihilation from the thermal particles
+takes over. In the light regime, however, when the mediator has a strong connection with
+the SM particles, the mediator reaches thermal equilibrium and acquires non-negligible
+corrections from the SM plasma. The thermal corrections modify the dispersion relation
+of the mediator, resulting in temperature-dependent mass effects.
+If the mediator is heavier than the dark sector, such a temperature-dependent mass ef-
+fect is expected to give a subdominant correction to the zero-temperature decay rate. How-
+ever, when the mediator is much lighter than the dark sector, the temperature-dependent
+mass enables a purely plasma-induced decay which is kinematically forbidden in vacuum.
+In this case, it is found that the rates from the scattering/annihilation of thermal parti-
+cles and the forbidden mediator decay are at the same order of coupling constants [22].
+Therefore, a consistent treatment from both scattering and the forbidden decay channels
+is needed to obtain a precise DM relic density.
+In the freeze-in paradigm [6, 7, 23–25] of nonthermal DM production, the contribution
+of the forbidden decay channel was considered by several studies [4, 10, 22, 26]. For the
+scalar mediator, the spectral density that encapsulates the thermal corrections at finite
+temperatures is usually a scalar function [27] in the Hard-Thermal-Loop approximation [28–
+32].
+For fermion mediators, however, the spectral density is more involved due to the
+helicity structure [33–35]. In computing the forbidden decay rate, the nontrivial helicity
+structure can cause significant difference between the vacuum tree-level amplitude and the
+thermal one-loop amplitude, as previously noticed in the applications to leptogenesis [36].
+The work aims to provide a dedicated study of freeze-in DM production via a thermal
+fermion mediator, where the mediator decay to DM is kinematically forbidden at zero
+temperature.
+We concentrate on the determination of the DM relic density from the
+forbidden decay and the scattering. We calculate the forbidden decay rates from a thermal
+one-loop amplitude and a vacuum tree-level amplitude, respectively, and find that the rate
+can be simply obtained from the latter with some constant rescaling. The comparison
+between the forbidden decay and the scattering shows a rather simple dependence on the
+thermal coupling constant, which enables us to include the plasma-induced decay in the
+scattering channel in an efficient way. This work complements the studies of nonthermal
+DM production through a light fermion mediator and provides a simple and comprehensive
+method to treat the forbidden decay for a wide range of fermion mediator scenarios.
+The remainder of this paper is outlined as follows. In Sec. 2, we present a simplified but
+general scenario to illustrate the freeze-in DM production via a light and thermal fermion
+mediator. Within the simplified scenario, we calculate the forbidden decay rate in Sec. 3
+and make a comparison with the rate derived from the vacuum tree-level amplitude. In
+Sec. 4, we first point out some subtleties concerning the double-counting issue and the
+s-channel resonant enhancement, and then evaluate the scattering rate without thermal
+corrections. In Sec. 5, we determine the DM relic density from the forbidden decay and
+– 2 –
+
+scattering channels respectively. More realistic scenarios based on Sec. 2 with potential
+observations will be discussed in Sec. 6. Conclusions are made in Sec. 7 and some technical
+details are relegated to the appendix.
+2
+The simplified scenario
+We first consider a simplified scenario in which the nonthermal dark sector consists of a
+Dirac fermion χ and a scalar φ.
+The connection between the dark sector and a Dirac
+fermion mediator ψ is realized by the following Yukawa interaction:
+LDM = yχ ¯ψRχLφ + h.c.
+(2.1)
+To ensure a thermal history of ψ, we consider a typical Yukawa interaction between the
+mediator and the thermal plasma, i.e.,
+Lψ = yψ ¯ψRηLϕ + h.c. ,
+(2.2)
+where both the fermion η and the scalar ϕ live in the thermal plasma. For clarity, we
+assume that the fermion mediator is right-handed in (2.1), but it should be mentioned that
+a left-handed fermion mediator is also possible. In Sec. 6, we shall discuss some realistic
+models for both right- and left-handed fermion mediators.
+Note that the fermion mediator can also have gauge interactions, e.g.,
+Vµ ¯ψRγµψR ,
+(2.3)
+with Vµ a U(1) gauge boson.
+Nevertheless, when the mediator is thermalized via the
+gauge interaction, gauge invariance requires that either χ or φ should be also charged
+under the gauge U(1) symmetry. In this case, either χ or φ will reach thermal equilibrium
+in the early universe, which can lead to significant difference from the situation where
+both χ and φ are far from equilibrium. For instance, when φ is in thermal equilibrium,
+the decay φ → χ + ψ and the scattering φ + ψ → χ + Vµ can dominate the production
+of χ, both of which are suppressed instead when φ is far from equilibrium. Besides, the
+Landau-Pomeranchuk-Migdal effect induced by soft vector boson exchange would also be of
+leading-order contribution [37] and should be taken into account consistently. Throughout
+this work, we will consider for simplicity a dark sector consisting of nonthermal χ and φ,
+leaving a thermal χ or φ for future studies.
+We will consider the situation where all the relevant thermal particles, i.e., ψ, η, and ϕ
+have vacuum masses much lighter than the dark sector, which is readily applicable to super-
+heavy DM [38, 39]. In this light regime, the freeze-in temperature of the DM is determined
+by the highest scale in the dark sector. Besides, the nonrelativistic annihilation of ψ, η, and
+ϕ to the dark sector is kinematically forbidden. Consequently, the DM relic density would
+basically be independent of the vacuum masses of the thermal particles. In the following
+discussions, we assume mχ < mφ for clarity. In this mass regime, either χ can be the only
+– 3 –
+
+χ
+χ
+ψ
+ϕ
++
++
+ϕ
+ψ
+χ
+χ
+−
+−
+Σχ
+−+
+Σχ
++−
+Figure 1. The one-loop self-energy diagrams of χ that contribute to the imaginary part of the
+retarded amplitude ImΣχ
+R in the forbidden decay. Here ± in the vertices denote the thermal indices
+in the doubled space of real-time formalism and the red blod denotes the resummed ψ propagator
+at finite temperatures.
+DM candidate or both χ and φ contribute to the observed DM relic density, though the
+later case is ruled out if mφ ≫ 1 GeV.
+3
+Forbidden decay
+3.1
+Boltzmann equation
+The decay process ψ → χ + φ is kinematically forbidden in vacuum but opened at finite
+temperatures.
+The forbidden decay rate that determines the density evolution in the
+dark sector can be calculated in the finite-temperature field theory [32]. Concerning the
+production of χ, the Boltzmann equation can be written as
+∂nχ
+∂t + 3Hnχ =
+�
+d3pχ
+(2π)3 (feq
+χ − fχ)Γχ ,
+(3.1)
+where feq
+χ (Eχ) = (eEχ/T + 1)−1 is the Fermi-Dirac distribution function of χ and H ≈
+1.66√gρT 2/MPl is the Hubble parameter with the effective number of relativistic degrees
+of freedom gρ for energy density and the Planck mass MPl ≈ 1.22 × 1019 GeV.
+The production rate Γχ at finite temperatures is related to the one-loop retarded self-
+energy of χ via [40]
+Γχ(P) = −gχ
+Tr[(/P + mχ)ImΣχ
+R(P)]
+2Ep
+,
+(3.2)
+with Pµ = (Ep, ⃗p) the 4-momentum of χ and ImΣχ
+R the imaginary part of the one-loop
+retarded amplitude. It should be mentioned that the factor of 2 in the denominator of
+Eq. (3.2) results from the spin sum and average over the Dirac spinor χ. Therefore, the
+collision rate in the Boltzmann Eq. (3.1) should be further multiplied by the spin degrees of
+freedom gχ = 2 [41] so as to obtain a collision term without spin average. For a nonthermal
+DM in the freeze-in paradigm, we expect fχ ≪ feq
+χ
+so that fχ can be neglected in the
+determination of the DM relic density. In the end, the relic density should be multiplied
+by a factor of 2 to take into account the antiparticle (¯χ) contribution.
+In the real-time formalism, the imaginary part of the retarded amplitude Σχ
+R can be
+– 4 –
+
+computed from the one-loop self-energy diagrams shown in Fig. 1, with
+ImΣχ
+R(P) = i
+2
+�
+Σχ
++−(P) − Σχ
+−+(P)
+�
+.
+(3.3)
+Using the expressions of Σχ
++−, Σχ
+−+ from Appendix A.1, we obtain
+ImΣχ
+R(P) =
+y2
+χ
+2(2π)2
+�
+d4Ksign(k0 − p0)fψ(k0)δ[(K − P)2 − m2
+φ]ρψ(K) ,
+(3.4)
+where sign(k0 − p0) denotes the sign function and fψ(k0) = (ek0/T + 1)−1. In the above
+equation, we have neglected the scalar distribution function fφ since φ is sparse during
+the freeze-in production.
+ρψ(K) is the spectral density that encapsulates the thermal
+corrections to ψ, as we shall derive below.
+3.2
+Spectral density of the fermion mediator
+The spectral density is defined via the resummed ψ propagators,
+S+− = −fψ( ˜GR − ˜GA) ≡ −2πifψ(k0)ρψ(K) ,
+(3.5)
+S−+ = [1 − fψ(k0)]( ˜GR − ˜GA) ≡ 2πi[1 − fψ(k0)]ρψ(K) ,
+(3.6)
+where ˜GR/ ˜GA are the resummed retarded/advanced propagators. Since the spectral den-
+sity defined above encapsulates the thermal corrections in the form of ˜GR − ˜GA, we should
+first be aware of how the thermal corrections appear in the resummed retarded and ad-
+vanced propagators.
+In general, the retarded amplitude for fermion self-energy can be parameterized as1 [33]
+−Σψ
+R(K) ≡ (aLPL + aRPR) /K + (bLPL + bRPR)/U ,
+(3.7)
+where PL,R are the chirality projection operators and Uµ is the four-velocity of the plasma
+with UµU µ = 1.
+In the rest frame, Uµ = (1, 0, 0, 0).
+Since the parity of the fermion
+mediator from the interactions given in Sec. 2 is explicitly broken, and at sufficiently high
+temperatures ψ is effectively massless2, we are essentially working in a chirality-symmetric
+and parity-broken theory, where aL, bL are nonzero while aR, bR = 0.
+The coefficients
+aL, bL can be calculated by left-multiplying Σψ
+R(K) with /K and /U, and then evaluating the
+trace. The general expressions read:
+aL =
+1
+2k2
+�
+Tr[ /KΣψ
+R(K)] − k0Tr[/UΣψ
+R(K)]
+�
+,
+(3.8)
+bL = − 1
+2k2
+�
+k0Tr[ /KΣψ
+R(K)] − K2Tr[/UΣψ
+R(K)]
+�
+,
+(3.9)
+with K2 = k2
+0 − k2.
+1The minus sign is defined for convenience, which results in 1 + a in the denominator of propagators.
+2If ψ acquires its vacuum mass via the Higgs or Higgs-like mechanism, then ψ is exactly massless above
+the cross-over or phase-transition temperature.
+– 5 –
+
+Given Eq. (3.7), the resummed retarded propagator in the chirality-symmetric and
+parity-broken regime can be written as
+˜GR = PR
+(1 + aL) /K + bL /U
+[(1 + aL)k0 + bL]2 − [(1 + aL)k]2 + isign(k0)ϵPL ,
+(3.10)
+and the advanced propagator can be similarly obtained by using Σψ
+A = Σψ∗
+R . The difference
+˜GR − ˜GA can be conveniently written in terms of the helicity eigenstates [34, 35],
+˜GR − ˜GA =
+�
+±
+−2i(Im∆+ ∓ sign(k0)ϵ)
+[Re∆±]2 + [Im∆± + ϵ]2 ˆP± ,
+(3.11)
+where ∆±(K) ≡ (1 + aL)k0 + bL ± (1 + aL)k, and the helicity operators are defined by
+ˆP± ≡ PR
+γ0 ± ⃗ek · ⃗γ
+2
+PL ,
+(3.12)
+with ⃗ek ≡ ⃗k/k.
+The spectral density ρψ can be decomposed into the on-shell and off-shell parts,
+ρψ(K) ≡ ρψ,on(K) + ρψ,off(K) .
+(3.13)
+The kinematically forbidden decay stems from the on-shell part ρψ,on(K), as will be derived
+in this section, while the off-shell part ρψ,off(K) arises from nonzero Im∆± and corresponds
+to the scattering channels. Note that the on-shell propagation of the fermion mediator
+could also result from the scattering channel. To avoid potential double counting, ρψ,on(K)
+defined above corresponds to Im∆± = 0. Then, from Eq. (3.11) the on-shell part is given
+by
+ρψ,on(K) =
+�
+±
+±sign(k0)
+���∂Re∆±
+∂k0
+���
+−1�
+δ(k0 − ω±
+1 ) + δ(k0 − ω±
+2 )
+�
+ˆP± .
+(3.14)
+In general, there are two solutions ω1,2 to Re∆i = 0 for each helicity operator ˆPi. In the
+free limit, aL = bL = 0 and ∆± = k0 ± k. It can be verified that S+−, S−+ given in
+Eqs. (3.5) and (3.6) reduce to the known forms [32]:
+S+−(K) = 2πisign(k0)fψ(k0)δ(K2) /K ,
+(3.15)
+S−+(K) = −2πisign(k0)[1 − fψ(k0)]δ(K2) /K .
+(3.16)
+To proceed with Eq. (3.4), the remaining task is to evaluate the real part of the resummed
+amplitude Σψ
+R, which depends on the thermal interaction specified in Sec. 2.
+The one-loop retarded self-energy diagram of ψ from (2.2) is similar to Fig. 1, with the
+resummed fermion propagators replaced by the free ones given in Eqs. (3.15) and (3.16).
+The inclusion of resummed propagators for the thermal η and ϕ in Fig. 1 is of higher order
+under the perturbative HTL technique. Substituting Eqs. (A.7) and (A.8) into Eqs. (3.8)
+– 6 –
+
+and (3.9), we obtain the real part of the coefficients aL, bL as
+ReaL =
+m2
+ψ(T)
+k2
+�
+1 + k0
+2k ln
+����
+k0 − k
+k0 + k
+����
+�
+,
+(3.17)
+RebL = −
+m2
+ψ(T)
+k
+�k0
+k − 1
+2
+�
+1 − k2
+0
+k2
+�
+ln
+����
+k0 − k
+k0 + k
+����
+�
+,
+(3.18)
+where the thermal mass is defined by
+m2
+ψ(T) =
+y2
+ψ
+16T 2 ≡ κ2T 2 .
+(3.19)
+Note that the scalar ϕ and fermion η can be gauge multiplets. For instance, if they are gauge
+SU(2) doublets, then an additional factor of 2 arises in m2
+ψ(T). We will not distinguish
+such difference but use κ as a free thermal parameter in later analyses.
+The results given in Eqs. (3.17) and (3.18) are consistent with Ref. [33] except that
+the logarithmic function is expressed by the modulus of momentum. The modulus arises
+when we integrate cos θ in Eq. (A.8) without restricting ourselves to the timelike regime
+K2 = k2
+0 −k2 > 0. Nevertheless, we will see below that an on-shell fermion with Eqs. (3.17)
+and (3.18) cannot propagate in the spacelike region. The modified dispersion relation is
+given by
+[(1 + ReaL)k0 + RebL]2 − [(1 + ReaL)k]2 = 0 .
+(3.20)
+For a weak-coupling theory yψ ≲ 1, we expect ReaL < 1. Neglecting the higher-order
+terms Rea2
+L and Reb2
+L, we obtain the approximate dispersion relation:
+k2
+0 − k2 ≈ − 2k0RebL
+1 + 2ReaL
+.
+(3.21)
+Then given Eqs. (3.17) and (3.18), it is straightforward to verify that there is no solution
+to the above equation for k2
+0 − k2 < 0. Therefore, the absolute symbol in Eqs. (3.17) and
+(3.18) should be removed.
+3.3
+Collision rate
+3.3.1
+One-loop retarded amplitude
+Given the expressions of ReaL, RebL in Eqs. (3.17) and (3.18), the on-shell spectral density
+from Eq. (3.14) can be simplified as
+ρψ,on(K) =
+�
+±
+± k2
+0 − k2
+2m2
+ψ(T)sign(k0) [δ(k0 ∓ ω1) + δ(k0 ± ω2)] ˆP± ,
+(3.22)
+– 7 –
+
+0.001
+0.005
+0.010
+0.050
+0.100
+0.005
+0.010
+0.020
+0.050
+0.01
+0.02
+0.03
+0.04
+0.05
+0.01
+0.02
+0.03
+0.04
+0.05
+Figure 2. The behavior of dispersion relation (3.20) for k/T ≪ 1, where the thermal parameter is
+set by κ = 0.01.
+where ω1,2 are the solutions to the modified dispersion relation (3.20) and can be analyti-
+cally expressed in terms of the Lambert W-function [36]:
+ω1 = −kW0(−e−2k2/m2
+ψ−1) − 1
+W0(−e−2k2/m2
+ψ−1) + 1
+,
+ω2 = kW−1(−e−2k2/m2
+ψ−1) − 1
+W−1(−e−2k2/m2
+ψ−1) + 1
+,
+(3.23)
+with ω1,2 > k.
+Substituting Eqs. (3.4) and (3.2) into the collision term in Eq. (3.1), we arrive at the
+decay rate
+Cχ,dec =
+y2
+χ
+32π3m2
+ψ(T)
+� ∞
+mχ
+dp0feq
+χ (p0)
+×
+� ∞
+0
+dk
+�
+i=1,2
+∓Θi(ω2
+i − k2)fψ(ωi)(±k2 ∓ ω2
+i + 2p0(k ± ωi) ∓ δm2) ,
+(3.24)
+where δm2 ≡ m2
+χ − m2
+φ < 0 and the symbol Θi imposes a restriction on the momentum
+integration from Eq. (3.4). Integrating the angle via the Dirac δ-function δ[(K −P)2 −m2
+φ]
+in Eq. (3.4), we find that in the timelike region K2 > 0 the restriction turns out to be
+K2 + δm2
+2(k0 + k) < p0 < K2 + δm2
+2(k0 − k) ,
+k0 − p0 > 0 .
+(3.25)
+Therefore, Θi is given by the Heaviside θ-function with
+Θi = θ
+�
+(2p0k)2 − (ω2
+i − k2 + δm2 − 2p0ωi)2�
+.
+(3.26)
+The solutions ω1,2 from the modified dispersion relation are shown in Figs. 2 and 3
+for k/T ≪ 1 and k/T > 1, respectively. It can been seen that when k becomes larger,
+the ω1-mode approaches a dispersion relation ω1 ≈ k while the ω2-mode approaches a
+– 8 –
+
+1.0000
+1.0002
+1.0004
+1.0006
+1.0008
+1.0010
+1.0000
+1.0002
+1.0005
+1.0008
+1.0010
+Figure 3. The behavior of dispersion relation (3.20) for k/T > 1.
+vacuum-like dispersion relation with an asymptotic mass
+√
+2mψ(T) [36, 42–44]. It allows
+us to compute Eq. (3.24) with the following approximations:
+ω2
+1 − k2 ≈ 0 ,
+ω2
+2 − k2 ≈ 2m2
+ψ(T) ,
+(3.27)
+which gives rise to Cχ,dec as
+Cχ,dec ≈
+y2
+χ
+16π3
+� ∞
+mχ
+dp0feq
+χ (p0)
+� ∞
+0
+dkΘ2fψ(ω2)
+�
+2m2
+ψ(T) + 2p0(k − ω2) + δm2�
+,
+(3.28)
+3.3.2
+Tree-level amplitude
+To see whether we can directly use the vacuum tree-level amplitude to compute the collision
+rate with the fermion thermal mass put in by hand, let us now calculate the relevant tree-
+level amplitude. As can be seen from Figs. 2 and 3, the ω1-mode quickly turns massless
+while the ω2-mode has an asymptotic mass
+√
+2mψ(T) so that sufficient momentum space
+is opened in this mode for the forbidden decay. In the following, we will use the dispersion
+relation ω2 − k2 = 2m2
+ψ(T) to calculate the decay rate from the tree-level amplitude.
+The squared amplitude of ψ → χ + φ is given by
+�
+s
+|M|2 ≈ y2
+χ(2κ2T 2 − m2
+φ) ,
+(3.29)
+where the approximation is obtained in the limit mχ ≪ mφ.
+Note that the squared
+amplitude for the dispersion relation ω2 −k2 = m2
+ψ(T) can be simply obtained by replacing
+√
+2κ with κ.
+– 9 –
+
+0.001
+0.010
+0.100
+1
+10-10
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+0.001
+0.010
+0.100
+1
+1
+2
+3
+4
+5
+6
+Figure 4. The comparison of forbidden decay rates from the one-loop retarded and vacuum tree-
+level amplitudes. Here ˜Cχ,dec ≡ y−2
+χ T −4Cχ,dec. In the weak-coupling regime κ < 1, the rates from
+the tree-level amplitude are overestimated by a factor of 1–4.
+The collision rate is given by
+Cχ,dec =
+�
+d3pψ
+(2π)32Eψ
+feq
+ψ
+�
+d3pχ
+(2π)32Eχ
+d3pφ
+(2π)32Eφ
+(2π)4δ4(Pψ − Pχ − Pφ)
+�
+s
+|M|2
+ψ→χφ
+≈ y2
+χκ3K1(
+√
+2κ)
+8
+√
+2π3
+�
+1 −
+m2
+φ
+2κ2T 2
+�2
+T 4 ,
+(3.30)
+where K1 is the modified Bessel function with K1(x) ≈ 1/x for x ≪ 1.
+In the last
+approximation we have used the Boltzmann distribution fψ(Eψ) = e−Eψ/T and kept the
+highest scale mφ from the dark sector.
+In the left panel of Fig. 4, we compare the decay rates obtained from Eq. (3.28) and
+Eq. (3.30) with different thermal parameter κ. Note that the rates from the two approaches
+share the same critical temperature
+Tc ≈ mφ
+√
+2κ,
+(3.31)
+after which the decay is kinematically closed. We can see that the rate from the tree-level
+amplitude with an effective mass
+√
+2mψ(T) is overestimated with respect to that from the
+one-loop retarded amplitude.
+In the right panel of Fig. 4, we also show the ratios of various decay rates by evaluating
+the vacuum tree-level amplitude with an effective mass mψ(T) and taking the full Fermi-
+Dirac statistics for feq
+ψ . Noticeably, a larger discrepancy between the retarded rate CR
+χ,dec
+and the vacuum one appears when the tree-level amplitude is evaluated with the asymptotic
+mass
+√
+2mψ(T), as seen from the C
+√
+2FD
+χ,dec /CR
+χ,dec and C
+√
+2MB
+χ,dec /CR
+χ,dec curves. Instead, the
+vacuum rates with the dispersion relation ω2 − k2 = m2
+ψ(T) are more compatible with the
+– 10 –
+
+retarded one. We found that for κ ≪ 1 the ratios reach
+CFD
+χ,dec
+CR
+χ,dec
+≈ 1.44 ,
+CMB
+χ,dec
+CR
+χ,dec
+≈ 1.75 ,
+(3.32)
+in which CFD
+χ,dec and CMB
+χ,dec denote the vacuum rates with the Fermi-Dirac and Maxwell-
+Boltzman statistics, respectively, together with the dispersion relation ω2−k2 = m2
+ψ(T). In
+particular, a smaller discrepancy can be seen between CFD
+χ,dec and CR
+χ,dec, since the latter is
+also derived from the full Fermi-Dirac statistics. It points out that the decay rate from the
+tree-level amplitude can coincide with that from the one-loop retarded amplitude within a
+factor of 2 in the generically weak-coupling regime κ < 1, if ω2 − k2 = m2
+ψ(T) is put in by
+hand in the tree-level amplitude.
+Since the ratios shown in the right panel of Fig. 4 are predicted via a common thermal
+parameter κ, and the ratios become nearly constant when κ ≲ 0.13 , the forbidden fermion
+decay rate can then be simply obtained from the tree-level amplitude with the approximate
+dispersion relation ω2−k2 ≈ m2
+ψ(T) and rescaling the latter by a factor of 0.69 if the Fermi-
+Dirac statistics is used or a factor of 0.57 if the Boltzmann distribution is used. It enables
+us to obtain a precise forbidden fermion decay rate within the simple tree-level approach
+by some constant rescaling.
+4
+Scattering
+4.1
+Double counting and resonant enhancement
+The scattering rate directly calculated from Fig. 1 is much more involved. The imaginary
+parts Im∆± appear both in the numerator and denominator of the off-shell spectral den-
+sity ρψ,off, making the final three-dimensional integration (dpdk0dk) difficult even with a
+numerical approach. For most situations, the thermal corrections to the scattering pro-
+cesses are significant only when there are IR singularities or resonance. For example, the
+IR singularity is known in neutrino and electron chirality-flipping processes at finite tem-
+peratures [41, 45–47], and the resonant effect from thermal corrections is also known in
+neutrino oscillations at finite temperature and density [48, 49].
+In dealing with the IR singularity or resonance, we can also use a more convenient
+approach in which the cross section is calculated from a tree-level diagram with a resummed
+mediator propagator [37, 50, 51]. When applying the effective approach, however, we should
+take care of the double-counting issue. There are in general two methods to remove the
+double counting. When the full thermal width of the mediator propagator is unknown,
+it is convenient to subtract the on-shell point directly from the cross section, and then
+calculate the forbidden decay rate separately. On the other hand, if the thermal width is
+known in a given model, a modified Breit-Wigner approximation can be applied to do the
+subtraction [52, 53], where the decay is automatically included in the cross section.
+Nevertheless, the double-counting issue depends on the existence of the resonance,
+which requires a careful inspection under the perturbative HTL resummation.
+In the
+3This corresponds to a generically weak-coupling regime yψ < 1.
+– 11 –
+
+following, let us concentrate on the s-channel double counting and on the hard particle
+scattering with incoming momenta phard ∼ O(T).
+Generically, hard scattering suffices
+to be responsible for the nonthermal DM production from thermal particles, since the
+thermally averaged collision rate ⟨σv⟩n is proportional to the particle-number densities of
+incoming thermal particles, which are expected to be dominated in the hard-momentum
+regime:
+nsoft ∝
+� psoft
+0
+d3pfeq(p) ∼ p3
+soft,
+nhard ∝
+� ∞
+psoft
+d3pfeq(p) ∼ T 3 ≫ p3
+soft ,
+(4.1)
+with psoft ∼ O(κT).
+At leading order, the mediator is resummed while the external particles are treated
+effectively massless. At this order, it is usually expected to have an s-channel resonance
+when the momentum transfer is near the scale of the effective mediator mass. However,
+when we go beyond the leading order, the external particles are resummed, which also carry
+effective masses from the plasma. If the thermal masses from the external particles are
+larger than from the mediator, the resonance expected at leading order would be erased.
+This is interpreted as the fact that the inverse decay X + Y → Z is always kinematically
+forbidden at all temperatures. This is particularly the case when the mediator is a fermion
+and the incoming particles contain a scalar boson. For instance, the resummed scalar ϕ
+has a thermal correction parameter κ = yψ/
+√
+12 [22] from the ψ − η loop, which is larger
+than the value given in Eq. (3.19).
+The above conclusion differs from two fermion scattering mediated by a thermal scalar.
+As seen from Figs. 2 and 3, there is a nearly massless state for a resummed fermion so
+that the initial fermions can have an approximate dispersion relation ω2
+i − k2 ≈ 0 while
+the resummed scalar mediator carries a large thermal mass. When s = k2
+0 − k2 ∼ κ2T 2,
+there is in principle an on-shell crossing and including the resummed scalar mediator in
+the fermion-pair scattering can enhance the scattering rate by a factor of O(1) [22].
+Since in current scenario the initial particles contain a fermion and a scalar boson,
+it is not necessary to use the resummed fermion mediator and the scattering rate from a
+vacuum computation suffices to describe the DM production to a good approximation.
+4.2
+Tree-level scattering amplitude without thermal correction
+The general 2 → 2 scattering rate for the DM production is given by
+C12→χφ =
+�
+d3p1
+(2π)32E1
+d3p2
+(2π)32E2
+d3pχ
+(2π)32Eχ
+d3pφ
+(2π)32Eφ
+f1f2|M|2
+12→χφ(2π)4δ4 ,
+(4.2)
+where δ4 ≡ δ4(P1 + P2 − Pχ − Pφ) and |M|2
+12→χφ is the squared amplitude with spin sum
+but without spin average. The Pauli blocking and Bose enhancement from the nonthermal
+DM sector are neglected.
+For Yukawa interaction, the scattering is η + ϕ → χ + φ. The squared amplitude is
+– 12 –
+
+0.001
+0.010
+0.100
+1
+10-10
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+Figure 5. A comparison between the forbidden decay and scattering rates for different thermal
+parameter κ. Here ˜Cχ ≡ y−2
+χ T −4Cχ.
+given by
+�
+s
+|M|2
+ϕη→χφ ≈
+y2
+χy2
+ψ
+2
+(1 −
+m2
+φ
+s )(1 + cos θ) ,
+(4.3)
+where we have only kept the highest mass scale from mφ and θ is the angle between
+the spatial momenta of the incoming and outgoing particles in the center-of-mass frame.
+Following the conventional phase-space reduction [54], we obtain the collision rate
+Cϕη→χφ =
+T
+32π4
+� ∞
+m2
+φ
+dsσϕη→χφs3/2K1(√s/T) ,
+(4.4)
+where the cross section without spin average is given by
+σϕη→χφ =
+y2
+χy2
+ψ
+32πs
+�
+1 −
+m2
+φ
+s
+�2
+.
+(4.5)
+In the high-temperature limit T ≫ mφ, the collision rate reduces to
+Cϕη→χφ ≈
+y2
+χy2
+ψ
+256π5 T 4 .
+(4.6)
+In Fig. 5, we show the rates from the forbidden decay and scattering channels. In
+general, Cχ,dec is larger than Cχ,scat when T > Tc.
+Nevertheless, the duration of the
+forbidden decay is determined by the critical temperature Tc, while the scattering η + ϕ →
+χ + φ is sufficiently closed only after the freeze-in temperature T ∼ mφ > Tc for κ < 1. It
+makes the scattering contribution to the final DM relic density generically larger than the
+forbidden decay, as we shall discuss below.
+– 13 –
+
+5
+DM relic density
+There are in principle two possibilities for DM relic density. If the scalar φ is unstable, it
+can decay to χ at late times after the dark sector freezes in. Consider first the situation
+where φ has been depleted away. χ is the DM candidate and the relic density is given by
+ΩDMh2 = (Y I
+χ + Y II
+χ )s0mχ
+ρc/h2
+.
+(5.1)
+where Y I
+χ ≡ nI
+χ/sSM is the yield produced by forbidden decay and scattering while Y II
+χ is
+the yield produced by scalar decay φ → ψ + χ at late times. sSM = 2π2gsT 3/45 is the
+SM entropy density with gs the effective number of relativistic degrees of freedom. The
+current value of entropy density is given by s0 = 2891.2 cm−3 and the current critical
+energy density ρc is given by ρc = 1.05 × 10−5 h2 · GeV · cm−3 [55].
+The Boltzmann equation for Y I
+χ is given by
+Y I
+χ =
+� ∞
+Tc
+2Cχ,dec
+sSMHT dT +
+� ∞
+0
+2Cχ,scat
+sSMHT dT ,
+(5.2)
+where the factor of 2 accounts for the CP-conjugated production so that Yχ is the sum
+of χ + ¯χ.
+The forbidden decay ends at T = Tc while the scattering basically ends at
+T = O(mφ) as the freeze-in temperature is determined by the highest scale in the dark
+sector. In the second term of Eq. (5.2), we use T = 0 as the lower integration limit, which
+does not cause significant difference after T drops below mφ/5. Since both χ + ¯χ and φ
+are produced with the same amount from the forbidden decay and scattering, we have
+Y I
+χ = Y I
+φ. Further given that the amount of χ + ¯χ in late-time production is inherited from
+Y I
+φ, we have Y II
+χ = Y I
+φ.
+Consider the second possibility where φ is sufficiently long-lived so that it has a lifetime
+comparable with or longer than the age of the observed universe. The DM relic density in
+this case consists of φ and χ, which is given by
+ΩDMh2 =
+s0
+ρc/h2 (Y I
+χmχ + Y I
+φmφ) .
+(5.3)
+To see the relative effect of the forbidden decay and the scattering channel, we estimate
+the ratio Yχ,scat/Yχ,dec, which reads:
+Yχ,scat
+Yχ,dec
+≈
+� xφ,fi
+0
+˜Cχ,scatdxφ
+� √
+2κ
+0
+˜Cχ,decdxφ
+,
+(5.4)
+where xφ ≡ mφ/T with xφ,fi corresponding to the freeze-in temperature. The evolution of
+˜Cχ,dec and ˜Cχ,scat can be found in Fig. 5. Simply taking ˜Cχ,dec and ˜Cχ,scat as constants,
+we obtain Yχ,scat/Yχ,dec ∝ 1/κ. It points out that the DM relic density from the forbidden
+decay basically carries an additional power of κ higher than from the scattering channel,
+even though both the decay and scattering rates share the same order of κ (see Eqs. (3.30)
+and (4.4)), as also found in Refs. [22, 26] in the case of forbidden scalar decay. The behavior
+– 14 –
+
+0.01
+0.05
+0.10
+0.50
+1
+1
+5
+10
+50
+0.01
+0.05
+0.10
+0.50
+1
+10-11
+10-10
+10-9
+10-8
+Figure 6.
+Left: A comparison of DM relic densities from the forbidden decay and scattering
+channels. Right: The correlation between the DM coupling yχ and the thermal parameter κ for the
+observed DM relic density. Here xD ≡ mχ/mφ.
+of Eq. (5.4) is shown in the left panel of Fig. 6 as a function of the thermal parameter κ.
+Note that only the highest scale mφ is kept in the yield so that both Yχ,dec and Yχ,scat are
+proportional to the inverse scalar mass, as expected from the IR freeze-in mechanism. We
+can see from the left panel of Fig. 6 that for the fermion mediator the forbidden decay can
+only be neglected for a very small κ. For a generically weak coupling 0.1 < yψ < 1, κ can
+reach O(0.1). For instance, about 41% of the DM relic density from Eq. (5.1) comes from
+the forbidden decay if κ = 0.5, while about 8% of the DM relic density is obtained from
+the forbidden decay if κ = 0.05.
+An interesting feature from such a comparison is that we can estimate the effect of the
+forbidden decay by rescaling the scattering rate, since the ratio given in Eq. (5.4) basically
+depends on the thermal coupling κ, or the interaction coupling yψ.
+Once the thermal
+interaction of the fermion mediator is known, we can calculate the scattering rate and
+simply rescale it by a κ- or yψ-dependent factor to obtain the forbidden decay. As shown
+in the left panel of Fig. 6, when κ ≲ 0.2, the ratio is approximately given by 0.56/κ and
+the total DM relic density given in Eq. (5.1) can then be estimated by
+ΩDMh2 ≈ 2 s0mχ
+ρc/h2 (1 + 1.79κ)Yχ,scat ,
+(5.5)
+where Yχ,scat comes from the second term in Eq. (5.2).
+In the right panel of Fig. 6, we plot the correlation between the DM coupling yχ and
+the thermal parameter κ by ��tting the observed DM relic density ΩDMh2 = 0.12 [56].
+The long-lived line corresponds to the second possibility from Eq. (5.3), where we have
+neglected the contribution from the light χ. In this approximation, the DM relic density
+is independent of mφ since Y I
+φ ∝ m−1
+φ .
+However, the DM relic density from Eq. (5.3)
+requires that the scalar should have a lifetime longer than the age of the universe, which is
+translated into an upper limit of the DM coupling yχ ≲ 10−20(mφ/GeV)−1/2. Therefore,
+we can conclude from the right panel of Fig. 6 that for a dark scalar heavier than 1 GeV,
+– 15 –
+
+the DM relic density from Eq. (5.3) is ruled out and the DM candidate can only be the
+lighter fermion χ. For instance, with yχ ≃ 10−11 and mφ ≃ 10 GeV, the scalar lifetime is
+around τφ ≃ 0.03 s. Thus the unstable heavy scalar has decayed away well before the BBN
+epoch.
+For the short-lived case from Eq. (5.1), the DM relic density depends on yχ, κ and
+the mass ratio in the dark sector xD ≡ mχ/mφ. We show in the right panel of Fig. 6 for
+three representative values xD = 0.1, 0.01, 0.001. We can see that when the mass ratio xD
+and the thermal parameter κ decrease, a larger DM coupling yχ is required to match the
+relic density. However, a large DM coupling could make the dark sector thermalized. To
+check this, recall that the nonthermal condition, which requires that the thermally averaged
+scattering rate should be smaller than the Hubble parameter at the freeze-in temperature,
+is given by
+Cχ,scat
+neq
+χ
+< H ,
+(5.6)
+where neq
+χ ≈ 0.09T 3 denotes the thermal particle-number density of χ. The above condition
+can be translated into an upper limit of the DM coupling yχ ≲ O(10−4). Therefore, for the
+thermal parameter κ and the mass ratio xD shown in the right panel of Fig. 6, the dark
+sector is indeed far from thermal equilibrium.
+When κ is much smaller but still able to keep the fermion mediator in thermal equi-
+librium, the scattering channel for the DM production can also come from the mediator
+scattering/annihilation, e.g., ψ + ¯ψ → χ + ¯χ mediated by the scalar φ and ψ + ¯ψ → φ + φ
+mediated by χ, both of which are not included in previous calculations since we are con-
+cerned with a relatively large κ. These scattering channels have rates at O(y4
+χ) and could
+be comparable with the thermal particle scattering ∼ O(y2
+χκ2) if yχ ∼ κ. For example,
+when the fermion mediator ψ is a GeV-scale right-handed neutrino in the type-I seesaw
+framework, the scattering ψ + ¯ψ → χ + ¯χ that can generate the observed DM relic den-
+sity predicts a nonthermal DM coupling yχ ∼ O(10−6) while the coupling for a GeV-scale
+right-handed neutrino to keep in thermal equilibrium via neutrino oscillation is required
+to be yψ > O(10−8) [21, 57]. Therefore, for a much smaller thermal parameter κ, the DM
+production from the mediator scattering/annihilation could be significant. A large thermal
+parameter κ, on the other hand, is usually more favorable as the strong connection between
+the SM and the fermion mediator enables us to have more opportunities of DM detection
+via the very fermion messenger.
+6
+Realistic scenarios and possible signals
+We have considered a simplified scenario in Sec. 2 where the nonthermal dark sector couples
+to the fermion mediator via the Yukawa interaction, and the thermal interaction for the
+mediator comes from chiral Yukawa interaction.
+In this section, we shall discuss some
+realistic models to which previous calculations can be applied.
+Right-handed Majorana/Dirac neutrino mediator.— Presumably, the most known
+example is the Majorana neutrino portal DM [11–21]. The left-handed fermion and the
+– 16 –
+
+scalar in Yukawa interaction (2.2) are specified as the SM lepton L and Higgs H doublets,
+respectively. Note that in this case, a light right-handed Majorana neutrino below the
+electroweak scale can readily be in thermal equilibrium via neutrino oscillation [21, 57].
+However, if the active-sterile neutrino mixing is small, the thermal corrections to the Ma-
+jorana neutrino would be suppressed. Consequently, the duration of the forbidden decay
+channel would be quite short and the scattering becomes the dominant channel to generate
+the DM relic density.
+ψR can also be specified as the right-handed Dirac counterpart of the SM left-handed
+neutrinos.
+The right-handed Dirac neutrinos can establish thermal equilibrium in the
+early universe via strong Yukawa interaction [58–60]. A noticeable difference between the
+Majorana and Dirac portals is that the later naturally predicts a very light fermion mediator
+with mass readily well below the dark scale.
+Both the Majorana and Dirac neutrino mediators naturally allow a dark sector to be
+produced via the freeze-in mechanism, as long as ψR does not have strong gauge inter-
+actions.
+In essence, the portal is realized by adding a SM gauge singlet to the super-
+renormalizable term ¯LH.
+When H is the SM Higgs doublet, the right-handed Majo-
+rana/Dirac neutrino portals naturally arise. It is also feasible that H is a non-SM scalar
+doublet and develops a vanishing vacuum expectation value. In this case, ψR is a more gen-
+eral neutral lepton singlet if there is no mass mixing between ψR and the SM left-handed
+neutrinos.
+Left-handed fermion mediator.— A left-handed fermion mediator can also couple to
+χR via chiral Yukawa interaction. For a nonthermal dark sector via the Yukawa interaction
+¯ψLχRφ, the left-handed mediator cannot have strong gauge interaction. There are some
+possibilities. For instance, ψL can couple to the SM charged-lepton singlet ℓR via
+yψ,i ¯ψLℓi,Rϕ + h.c. ,
+(6.1)
+where yψ,i in general have three couplings to the charged-lepton flavors, ψL is a neutral
+lepton and ϕ is electrically charged. Here ψ is a SM singlet so that the dark sector does
+not carry SM gauge charges. The thermalization of ψL can be easily realized if the above
+interaction is strong. Another possibility is that the charged-lepton singlet ℓR is replaced by
+the quark singlet. For instance, the down-quark singlet dR couples to ψL with a leptoquark
+scalar ϕ [61–65]
+¯dRψLϕ + h.c. ,
+(6.2)
+where the scalar ϕ is now an SU(3)c triplet and SU(2)L singlet, carrying the hypercharge
+Y = −1/3 so that ψ is a SM singlet.
+In all these cases, the fermion mediator can readily be thermalized in the SM thermal
+plasma. As seen from the right panel of Fig. 6, the connection between the thermal plasma
+and the fermion mediator will be enhanced if the coupling between the mediator and the
+dark sector is sufficiently small, and vice versa. In general, a smaller DM coupling makes
+the direct DM detection much more challenging but meanwhile the indirect signals from
+– 17 –
+
+the mediator may be boosted. On the other hand, the direct freeze-in DM direction may
+also be possible if the production cross section is enhanced e.g.
+by a sufficiently light
+mediator [66]. In the following, we shall discuss some possible signals that may be probed
+in current and future experiments.
+Cosmic flux from DM annihilation.— If the fermion mediator has a vacuum mass at
+GeV scale or above, the annihilation from DM to the mediator 2χ → 2ψ can potentially
+generate secondary fluxes consisting of SM particles via decay ψ → SM. For instance, the
+DM annihilation from the galactic center to right-handed Majorana neutrinos 2χ → 2νR
+can generate a secondary left-handed neutrino flux via active-sterile neutrino mixing [67,
+68]. For right-handed Dirac neutrino portal, it may also be interesting to consider the
+active neutrino flux from the DM annihilation 2χ → 2νR followed by a chirality-flipping
+process νR → νL caused e.g. by magnetic fields in the universe [69–71].
+Extra radiation in the early universe.— If the fermion mediator is sufficiently light,
+e.g., in the right-handed Dirac neutrino portal scnearios, the light mediator itself can
+significantly contribute to the energy density of the early universe, thereby leaving potential
+imprints in the BBN/CMB regimes. In particular, the light mediator may produce an Neff
+excess which can be probed in future experiments [72–74].
+LHC detection.— If the light fermion mediator is sufficiently long-lived, it can generate
+displaced vertices at the LHC [75–77], such as a long-lived right-handed neutrino [78] or
+a neutral ψL produced by the electron-positron pair in the t channel via Eq. (6.1). In the
+later case, if mψ > mµ + mϕ, the fermion mediator can decay into a charged scalar and
+a muon, leaving displaced vertices in a remote muon chamber if the decay length cτψ is
+sufficiently long. If mψ < mϕ, the opposite-sign dilepton can be produced with missing
+energy from the charged scalar decay ϕ± → ℓ± + ψ [79].
+7
+Conclusions
+In this work we have concentrated on the freeze-in DM production via a light fermion
+mediator once thermalized in the early universe. We have used a simplified scenario to
+capture the basic properties of such a class of DM models, which can be applied in the
+scenarios of right-handed Majarona/Dirac neutrino portals and the left-handed fermion
+mediator coupling to charged leptons and quarks.
+When the fermion mediator is much lighter than the dark sector, both the forbidden
+decay and the scattering should be taken into account consistently. The full forbidden
+decay rate is calculated from the one-loop retarded amplitude under the HTL approxi-
+mation at finite temperatures, which is always overestimated from a tree-level amplitude.
+Nevertheless, we found that the full forbidden decay can still be simply obtained from the
+tree-level amplitude after being rescaled by proper constants.
+While both the scattering and forbidden decay rates carry the same order of coupling
+constants, the scattering generically dominates the production as its duration in the pro-
+duction history is longer than in the forbidden decay. Nevertheless, the contribution from
+the forbidden decay is significant when the thermal interaction between the fermion medi-
+ator and the thermal plasma is strong. For a generically weak interaction that thermalizes
+– 18 –
+
+the light fermion mediator, the forbidden decay can contribute to the total DM relic den-
+sity at about 40%, and hence cannot be neglected in the precise calculation of DM relic
+density.
+Acknowledgments
+The author thanks Xun-Jie Xu for valuable discussions. This work is supported in part by
+the National Natural Science Foundation of China under grant No. 12141501.
+A
+Thermal one-loop amplitudes
+A.1
+The DM part
+The amplitudes from Fig. 1 are given by
+Σχ
++−(P) = −iy2
+χ
+�
+d4K
+(2π)4 G−+(K − P)S+−(K)
+=
+iy2
+χ
+(2π)2
+�
+d4Ksign(k0 − p0)[1 + fφ(k0 − p0)]fψ(k0)δK−P ρψ(K) ,
+(A.1)
+Σχ
+−+(P) = −iy2
+χ
+�
+d4K
+(2π)4 G+−(K − P)S−+(K)
+= −iy2
+χ
+(2π)2
+�
+d4Ksign(k0 − p0)fφ(k0)[1 − fψ(k0 − p0)]δK−P ρψ(K) ,
+(A.2)
+where δK−P ≡ δ[(K − P)2 − m2
+φ] and the free scalar propagators G−+, G+− are given by
+G+−(K) = −2πisign(k0)fφ(k0)δ(K2 − m2
+φ) ,
+(A.3)
+G−+(K) = −2πisign(k0)[1 + fφ(k0)]δ(K2 − m2
+φ) ,
+(A.4)
+while the resummed fermion propagators S+−, S−+ are given by Eqs. (3.5) and (3.6).
+A.2
+The fermion mediator part
+The real part of the retarded amplitude Σψ
+R(K) is equivalent to the time-ordered one
+Σψ
+++(K), which in the massless limit is given by
+Σψ
+++(K) = iy2
+ψ
+�
+d4Q
+(2π)4 G++(Q − K)PLS++(Q)PR
+= iy2
+ψ
+�
+d4Q
+(2π)4
+�
+1
+Q2 + iϵ + 2πifη(|q0|)δ(Q2)
+�
+PL /QPR
+×
+�
+1
+(Q − K)2 + iϵ − 2πifϕ(|q0 − k0|)δ[(Q − K)2]
+�
+,
+(A.5)
+– 19 –
+
+The zero-temperature part is UV divergent, which can be renormalized as usual in zero-
+temperature QFT. For the finite-temperature part, it reads
+ReΣψ
+R(K) =
+y2
+ψ
+(2π)3
+�
+d4Q
+�δ[(Q − K)2]
+Q2
+fϕ(|q0 − k0|) −
+δ(Q2)
+(Q − K)2 fη(|q0|)
+�
+PL /QPR
+=
+y2
+ψ
+(2π)3
+�
+d4Q
+δ(Q2)
+(Q − K)2
+�
+fϕ(q)PL(−/Q + /K)PR − fη(q)PL /QPR
+�
+,
+(A.6)
+where (Q − K)2 ̸= 0 and the second equation is obtained by replacing Q → −Q + K in the
+first term of the first equation. The above integration can be done as follows. Integrate
+q0 first via δ(Q2), then expand the denominator (Q − K)2 = K2 − 2K.Q in the HTL
+approximation: K2 ≪ q24, after that integrate the angle cos θ, and finally integrate the
+momentum q.
+In the HTL approximation, the trace given in Eqs. (3.8) and (3.9) are evaluated to be
+tr[ /KReΣψ
+R(K)] =
+2y2
+ψ
+(2π)2
+�
+q[fϕ(q) + fη(q)]dq + O(K2/q2)
+≈
+y2
+ψ
+8 T 2 ,
+(A.7)
+tr[/UReΣψ
+R(K)] =
+y2
+ψ
+(2π)2
+�
+q[fϕ(q) + fη(q)]dq
+�
+d cos θ
+k0
+k2
+0 − k2 cos θ2 + O(K2/q2)
+≈
+y2
+ψ
+16k ln
+����
+k0 + k
+k0 − k
+���� T 2 .
+(A.8)
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+arXiv:2301.03756v1  [math.PR]  10 Jan 2023
+Brownian Hitting to Spheres
+Yuji Hamana and Hiroyuki Matsumoto
+Abstract
+Let Sd−1
+r
+be the sphere in Rd whose center is the origin and the radius
+is r, and σr be the first hitting time to it of the standard Brownian motion
+{Bt}t≧0, possibly with constant drift. The aim of this article is to show
+explicit formulae by means of spherical harmonics for the density of the
+joint distribution of (σr, Bσr) and to study the asymptotic behavior of the
+distribution function. 1
+1.
+Introduction and main results
+For d ≧ 2, we consier a standard d-dimensional Brownian motion B = {Bt}t≧0
+starting from a fixed point (a, 0, ..., 0), where we assume a > 0, defined on a
+probability space (Ω, F, Pa). Letting Sd−1
+r
+be the sphere in Rd with radius r and
+centered at the origin, we are concerned with the joint distribution of the first
+hitting time σr of B to Sd−1
+r
+and the hitting place Bσr.
+The aim of this article is to show an explicit expression for the density of the
+joint distribution by means of the spherical harmonics, that is, the Gegenbauer
+and the Chebyshev polynomials. As an application, we study the asymptotic
+behavior of the tail probability Pa(t < σr < ∞, Bσr ∈ A), A ⊂ Sd−1
+b
+when a > r.
+The joint density for the Brownian motion with constsnt drift is also investigated.
+Several authors have studied the joint distribution. It should be first noted
+that in the exit problem, that is the case of a < r, the joint density is given
+by a solution for a heat equation with the Dirichlet boundary condition. See
+Aizenman-Simon [1] for general discussion and Hsu [9] for an explicit expression
+in the case of spheres. Wendel [19] has shown a nice result on the expectations
+of functions of (σb, Bσb) by using the spherical harmonics.
+See Gzyl [4] and
+references therein for a recent study on this direction and Uchiyama [17, 18] on
+the asymptotic behavior of the distribution functions and its application to the
+Wiener sausage. A similar problem for a Brownian motion with drift has been
+discussed in Yin-Wang [20].
+We proceed to a different way. Starting from the skew-product representation
+of Brownian motion, we use the fact due to Mijatovic-Mramor-Uribe Bravo [13]
+that the projections of the Brownian motion on the sphere Sd−1 = Sd−1
+1
+define
+diffusion processes. We see that, for one-dimensional projections, the eigenvalues
+12020 Mathematics Subject Classification: 60J65
+keywords : Brownian motion, hitting times and places, spherical harmonics, one-dimensional
+diffusion,
+1
+
+and the eigenfunctions for the generators are explicitly given by the spherical
+harmonics.
+Combining these facts with the rotation invariance of the probability law of
+Brownian motion, we show the following. As usual we denote by Iν and Kν the
+modified Bessel functions. We also denote by Cν
+n and Tn the Gegenbauer and the
+Chebyshev polynomials, respectively.
+Theorem 1.1. Denote by Ea the expectation with respect to Pa. Then, for λ > 0
+and u ∈ Rd, we have
+Ea[e−λσre⟨u,Bσr ⟩] =L0(a
+√
+2λ)
+L0(r
+√
+2λ)
+�
+S1 er⟨u,z⟩ds(z)
++ 2
+∞
+�
+n=1
+Ln(a
+√
+2λ)
+Ln(r
+√
+2λ)
+�
+S1 er⟨u,z⟩Tn(z1)ds(z)
+when d = 2 and
+Ea
+�
+e−λσre⟨u,Bσr ⟩I{σr<∞}
+�
+= 1
+ν
+∞
+�
+n=0
+(n + ν)a−νLn+ν(a
+√
+2λ)
+r−νLn+ν(r
+√
+2λ)
+�
+Sd−1 er⟨u,z⟩Cν
+n(z1)ds(z)
+when d ≧ 3, where ds is the uniform probability measure on Sd−1, and L = I for
+a < r and L = K for a > r.
+Setting u = 0 and noting that the surface integrals of Tn(z1) and Cν
+n(z1) vanish
+for n ≧ 1, we recover the well known formula for Ea[e−λσr] (cf. [2]).
+We can invert the joint Laplace transform and obtain the following. We denote
+by ρ(ν)
+a,r(t) the probability density of the first hitting time to r of a Bessel process
+with index ν starting from a.
+Theorem 1.2. For t > 0 and z ∈ Rd with |z| = r, we have
+Pa(σr ∈ dt, Bσr ∈ dz) = ρ(0)
+a,r(t)dtdsr(z) + 2
+∞
+�
+n=1
+�a
+r
+�nρ(n)
+a,r(t)Tn
+�z1
+r
+�
+dtdsr(z)
+when d = 2 and
+Pa(σr ∈ dt,Bσr ∈ dz)
+= 1
+ν
+∞
+�
+n=0
+�
+n + ν
+��a
+r
+�nρ(n+ν)
+a,r
+(t)Cν
+n
+�z1
+r
+�
+dtdsr(z)
+when d ≧ 3, where ν = d−2
+2
+and dsr is the uniform probability measire on Sd−1
+r
+.
+The authors [8] have shown another expression for the joint Laplace transform,
+from which we can prove Theorem 1.2.
+The rest of this article is organized as follows. In the next Section 2 we study
+the first coordinate or the one-dimensional projection of the Brownian motion
+on Sd−1.
+We give proofs of the theorems mentioned above in Section 3 and,
+the asymptotic behavior of Pa(t < σr < ∞, Bσr ∈ A), A ⊂ Sd−1
+r
+, as t → ∞
+is investigated in Section 4. In the final Section 5, we deal with the Brownian
+motion with constant drift.
+2
+
+2.
+Projection of Brownian motion on sphere
+Let θ = {θ(t)}t≧0 be a Brownian motion on Sd−1, which corresponds to the
+Laplace-Beltrami operator on Sd��1, endowed with the usual Euclidean metric.
+Mijatovic-Mramor-Uribe Bravo [13] has shown that the projections of θ are dif-
+fusion processes which are realized as unique solutions of stochastic differential
+equations. This fact, especially on the one-dimensional projections, is fundamen-
+tal in our argument and we recall the result in this special case.
+Proposition 2.1. The first coordinate {θ1(t)}t≧0 of θ is a diffusion process on
+(−1, 1) whose generator is
+Gd = 1
+2(1 − x2) d2
+dx2 − d − 1
+2
+x d
+dx.
+We see easily that the boundaries ±1 are regular and reflecting when d = 2
+and they are entrance ones when d ≧ 3. The eigenvalues and the eigenfunctions of
+Gd are explicitly given and we have the eigenfunction expansion for the transition
+densities.
+Since these play important roles in the following sections, we now
+recall some fundamental facts. For details of the Chebyshev and the Gegenbauer
+polynomials below, we refer to [3, 12, 14].
+Write
+Gd =
+1
+2(1 − x2)
+d−3
+2
+d
+dx
+�
+1
+(1 − x2)− d−1
+2
+d
+dx
+�
+and let dm(x) = 2(1 − x2)
+d−3
+2 dx be the canonical (speed) measure. Note that m
+is a finite measure on (−1, 1). Moreover, we take
+s(x) =
+� x
+0
+(1 − y2)− d−1
+2 dy
+as the scale function.
+When d = 2, s(±1) are both finite and the boundaries are regular.
+The
+Chebyshev polynomial Tn(x) = cos(n arccos x) satisfies
+G2Tn = −n2
+2 Tn
+and
+d
+dsTn(±1) = 0.
+Moreover the orthogonality relation is given by
+� 1
+−1
+Tm(x)Tn(x)
+dx
+√
+1 − x2 =
+
+
+
+
+
+0
+m ̸= n
+π
+2
+m = n ̸= 0
+π
+m = n = 0.
+Hence, setting
+φ0
+0(x) =
+1
+√
+2π,
+φ0
+n(x) =
+1
+√πTn(x)
+(n ≧ 1),
+3
+
+we see that {φ0
+n}∞
+n=0 gives rise to an orthonormal basis of L2(dm) and that the
+transition density p2(t, x, y) of {θ1(t)} with respect to dm is given by
+p2(t, x, y) = 1
+2π + 1
+π
+∞
+�
+n=1
+e− 1
+2n2tTn(x)Tn(y).
+(2.1)
+For d ≧ 3, the eigenfunctions are given by the Gegenbauer polynomials Cν
+n
+defined by
+∞
+�
+n=0
+snCν
+n(x) =
+1
+(1 + s2 − 2sx)ν ,
+|s| < 1,
+where ν = (d − 2)/2. In fact, we have
+GdCν
+n = −1
+2n(n + 2ν)Cν
+n
+and
+d
+dsCν
+n(±1) = 0
+and the orthogonality relation
+� 1
+−1
+Cν
+m(x)Cν
+n(x)(1 − x2)ν− 1
+2dx = δm,n
+πΓ(n + 2ν)
+22ν−1(n + ν)n!(Γ(ν))2.
+Hence, setting
+φν
+n(x) =
+� (n + ν)n!
+πΓ(n + 2ν)
+� 1
+22ν−1Γ(ν)Cν
+n(x),
+we obtain an orthonormal basis {φν
+n}∞
+n=0 of L2(dm) and an eigenfunction expsn-
+sion for the transition density pd(t, x, y) of {θ1(t)} with respect to dm,
+pd(t, x, y) =
+∞
+�
+n=0
+e− 1
+2n(n+2ν)tφν
+n(x)φν
+n(y).
+(2.2)
+3.
+Proof of Theorems 1.1 and 1.2
+We use the same notation as those in Section 1 and start the argument from
+the skew-product representation of the standard Brownian motion B = {Bt}t≧0:
+there exists a d-dimensional Bessel process R = {Rt}t≧0 (with index ν = (d−2)/2)
+and a Brownian motion θ = {θ(t)}t≧0 on Sd−1, independent of R, such that
+Bt = Rtθ(Ξt),
+Ξt =
+� t
+0
+ds
+R2s
+.
+B0 = (a, 0, ..., 0) means R0 = a and θ(0) = (1, 0, ..., 0). By the independence of
+R and θ, we have
+Ea[e−λσre⟨u,Bσr ⟩] = E(ν)
+a [e−λτrEa[er⟨u,θ(t)⟩]
+���
+t=Ξτr
+],
+where E(ν)
+a [ · ] denotes the expectation with respect to the probability law of R
+and τr is the first hitting time of R to r.
+4
+
+First we prove the theorems when d = 2. Writing θ(t) = (θ1(t), θ2(t)) and
+u = (u1, u2), we have by the rotation invariance of the law of standard Brownian
+motion
+Ea[er⟨u,θ(t)⟩] = Ea[eru1θ1(t)Ea[eru2θ2(t)|θ1(t)]]
+=
+� 1
+−1
+eru1y 1
+2
+�
+eru2√
+1−y2 + e−ru2√
+1−y2�
+P(θ1(t) ∈ dy).
+Hence formula (2.1) implies
+Ea[er⟨u,θ(t)⟩]
+= 1
+2π
+� 1
+−1
+eru1y 1
+2
+�
+eru2√
+1−y2 + e−ru2√
+1−y2�
+2dy
+�
+1 − y2
++ 1
+π
+∞
+�
+n=1
+e− 1
+2n2t
+� 1
+−1
+eru1y 1
+2
+�
+eru2√
+1−y2 + e−ru2√
+1−y2�
+Tn(y)
+2dy
+�
+1 − y2
+since Tn(1) = 1. The change of order of the intengal and the sum is easily justified
+because |Tn(y)| ≦ 1. We can write the integrals on the right hand side as surface
+integrals and obtain
+Ea[er⟨u,θ(t)⟩] =
+�
+S1 er⟨u,z⟩ds(z) + 2
+∞
+�
+n=1
+e− 1
+2n2t
+�
+S1 er⟨u,z⟩Tn(z1)ds(z).
+Now, recalling the formula ([2, p.407])
+E(0)
+a [e−λτr− 1
+2 n2Ξτr] = Ln(a
+√
+2λ)
+Ln(r
+√
+2λ)
+,
+(3.1)
+we obtain the assertion of Theorem 1.1 when d = 2.
+Next note another formula ([2, p.398])
+E(µ)
+a [e−λτr] =
+� ∞
+0
+e−λtρ(µ)
+a,r(t)dt = a−µLµ(a
+√
+2λ)
+r−µLµ(r
+√
+2λ)
+.
+Then we obtain Theorem 1.2 when d = 2. Again we can easily show the absolute
+convergence and justify the change of the sum and the integrals in t.
+Next we prove the theorems in the case of d ≧ 3, when, for the spherical
+Brownian motion θ, the conditional distribution of (θ2(t), ..., θd(t)) given θ1(t) =
+ξ1 is the uniform distribution on the sphere Sd−2
+√
+1−ξ2
+1 with raduis
+�
+1 − ξ2
+1. Hence,
+writing u = (u1, u′), θ = (θ1, θ′) ∈ R × Rd−1, we have
+Ea[e⟨u,rθ(t)⟩] = Ea
+�
+eru1θ1(t)
+�
+Sd−2 er√
+1−θ1(t)2⟨u′,ξ′⟩ ds(ξ′)
+�
+.
+5
+
+By using the facts on the Gegenbauer polynomials given in the previous section
+and writing the double integral as a surface integral, we obtain, from (2.2)
+Ea[e⟨u,rθ(t)⟩]
+=
+∞
+�
+n=0
+e− 1
+2n(n+2ν)tφν
+n(1)
+� 1
+−1
+φν
+n(ξ1)eru1ξ12(1 − ξ2
+1)
+d−3
+2 dξ1
+×
+�
+Sd−2 er√
+1−ξ2
+1⟨u′,ξ′⟩ vol(dξ′)
+vol(Sd−2)
+=
+∞
+�
+n=0
+(n + ν)22ν−1Γ(ν)2 vol(Sd−1)
+πΓ(2ν) vol(Sd−2)
+e− 1
+2 n(n+2ν)t
+�
+Sd−1 Cν
+n(w1)er⟨u,w⟩ds(w).
+We have used the formula Cν
+n(1) =
+�2ν+n−1
+n
+�
+= Γ(n + 2ν)/(n!Γ(2ν)), and also the
+estimate
+max
+|y|≦1 |Cν
+n(y)| = Cν
+n(1) ≦ Cn2ν−1
+(3.2)
+for some constant C (see, e.g., [12, pp.218, 225]) to justify the change of the order
+of the sum and the integration.
+Moreover, recalling the foumulae
+vol(Sd−1) = 2π
+d
+2
+Γ( d
+2)
+and
+Γ(2ν) = 22ν
+2√πΓ(ν)Γ(ν + 1
+2),
+we obtain
+Ea[e⟨u,rθ(t)⟩] = 1
+ν
+∞
+�
+n=0
+(n + ν)e− 1
+2n(n+2ν)t
+�
+Sd−1 Cν
+n(w1)er⟨u,w⟩ds(w).
+Now, using (3.1), we obtain the assertion of Theorem 1.1. Theorem 1.2 is proven
+in the same way as in the case of d = 2.
+4.
+Asymptotic behavior of distribution function
+In this section, assuming a > r and applying Theorem 1.2, we study the asymp-
+totic behavior of the distribution function Pa(t < σr < ∞, Bσr ∈ A) as t → ∞
+for a fixed Borel subset A of the sphere Sd−1
+r
+. We use the same notation as in the
+previous sections.
+In a course of study on the first hitting times of Bessel processes, the authors
+[6, 7] have shown the following.
+Consider a Bessel process with index ν and
+starting from a defined on some probability space (Ω′, F ′, Q(ν)
+a ) and let τr be its
+hitting time to r. Then the asymptotic behavior of Q(ν)
+a (t < τr < ∞) when a > r
+is given by
+Q(0)
+a (t < τr < ∞) = 2 log(a/r)
+log t
+(1 + o(1))
+(4.1)
+when d = 2 and
+Q(ν)
+a (t < τr < ∞) = κ(ν)t−ν(1 + o(1)),
+(4.2)
+6
+
+when d ≧ 3, where the constant κ(ν) is given by
+κ(ν) =
+1
+Γ(ν + 1)
+� r3
+2a
+��a
+r
+�ν
+−
+�a
+r
+�−ν�
+.
+Applying these results with some estimates for the remainder terms, we show
+the following.
+Theorem 4.1. For any Borel subset A of Sd−1
+r
+,
+Pa(t < σr < ∞, Bσr ∈ A) = 2 log(a/r)
+log t
+sr(A)(1 + o(1))
+holds as t → ∞ when d = 2 and
+Pa(t < σr < ∞, Bσr ∈ A) = κ(ν)sr(A)t−ν(1 + o(1))
+holds when d ≧ 3.
+Remark 4.2. For the distribition function Q(ν)
+a (t < τr < ∞) of the first hitting
+time of the Bessel process, Hamana et al. [5] has shown a precise asymptotic
+expansion and, using the results, we can show asymptotic expansions for our
+joint distribution functions. The details will be published elsewhere.
+For a proof of Theorem 4.1, we show the following estimate for the tail prob-
+ability of σr.
+Lemma 4.3. Assume d ≧ 3. Then, for t > 0, we have
+Pa(t < σr < ∞) ≦
+r2ν
+2νΓ(ν + 1)tν .
+Proof. Let Lr be the last hitting time of the Brownian motion B to the spehere
+Sd−1
+r
+:
+Lr = sup{s > 0 : |Bs| = r}.
+As usual we set Lr = 0 when B does not hit Sd−1
+r
+. Then we have
+Pa(t < σr < ∞) ≦ Pa(t < Lr < ∞).
+Denote by µr the equilibrium measure of the ball Br with radius r and centered
+at the origin. Then it is well known ([16]) that
+Pa(t < Lr < ∞) =
+� ∞
+t
+ds
+�
+Rd
+1
+(2πs)d/2e− |x−a|2
+2s dµr(x).
+Recalling now that the capacity of Br is µr(Rd) = 2π
+d
+2rd−2/Γ( d
+2 − 1), we see
+Pa(t < Lr < ∞) ≦
+� ∞
+t
+ds
+�
+Rd
+1
+(2πs)d/2dµr(x) =
+r2ν
+2νΓ(ν + 1)tν .
+7
+
+Remark 4.4. For transient one-dimensional diffusion processes, the densities of
+the last hitting times are written by means of the transition densities. This is
+the case of the Bessel processes with dimensions d ≧ 3 and, moreover, we have
+explicit expressions for the transition densities We can give another proof for
+Lemma 4.3 by using these facts.
+We can now give a proof of Theorem 4.1. Note that the infinite sum below
+for the expression for the joint distribution is absoletely convergent.
+For d = 2, we have by Theorem 1.2
+Pa(t < σr < ∞, Bσr ∈ A) = Q(0)
+a (τr > t)sb(A) + It,
+where
+It = 2
+∞
+�
+n=1
+�a
+r
+�nQ(n)
+a (t < τr < ∞)
+�
+A
+Tn(z1
+r )dsr(z).
+Assume t > 1 and note |Tn(x)| = | cos(n arccos x)| ≦ 1. Then, by Lemma 4.3, we
+get
+|It| ≦ 2
+∞
+�
+n=1
+�a
+r
+�n
+r2n
+2nΓ(n + 1)tn ≦ 2
+t
+∞
+�
+n=1
+�ar
+2
+�n 1
+n! ≦ 2
+t e
+ar
+2
+and, by (4.1), the assertion of Theorem 4.1.
+For d ≧ 3, we have
+Pa(t < σr < ∞, Bσr ∈ A) = Q(ν)
+a (t < τr < ∞)sr(A) + Jt,
+where
+Jt = 1
+ν
+∞
+�
+n=1
+(n + ν)
+�a
+r
+�nQ(n+ν)
+a
+(t < τr < ∞)
+�
+A
+Cν
+n(z1
+r )dsr(z).
+Hence, combining (3.2) with Lemma 4.3 and (4.2), we see Jt = O(t−1−ν) and the
+assertion of Theorem 4.1.
+5.
+Brownian motion with drift
+Let B = {Bt}t≧0 be a standard d-dimensional Brownian motion starting from
+x = (a, 0, ..., 0) as before and, for a constant vector v ∈ Rd, B(v) = {B(v)(t)}t≧0
+be a Brownian motion with drift v defined by B(v)(t) = Bt + tv. We denote by
+σ(v)
+r
+the first hitting time of B(v) to the sphere Sd−1
+r
+.
+The Cameron-Martin theorem and the strong Markov property of Brownian
+motion imply
+Ea
+�
+e−λσ(v)
+r e⟨u,B(v)(σ(v)
+r
+)⟩I{σ(v)
+r
+<∞}
+�
+= e−av1Ea
+�
+e−(λ+ |v|2
+2 )σre⟨u+v,Bσr ⟩I{σr<∞}
+�
+.
+Hence we can apply Theorem 1.1 to the right hand side and obtain the following:
+8
+
+Theorem 5.1. For λ > 0 and u ∈ Rd, we have
+Ea
+�
+e−λσ(v)
+r e⟨u,B(v)(σ(v)
+r
+)⟩I{σr<∞}
+�
+= e−av1
+�
+L0(a
+�
+2λ + |v|2)
+L0(r
+�
+2λ + |v|2)
+�
+S1 er⟨u+v,z⟩ds(z)
++ 2
+∞
+�
+n=1
+Ln(a
+�
+2λ + |v|2)
+Ln(r
+�
+2λ + |v|2)
+�
+S1 er⟨u+v,z⟩Tn(z1)ds(z)
+�
+when d = 2 and, when d ≧ 3,
+Ea
+�
+e−λσ(v)
+r e⟨u,B(v)(σ(v)
+r
+)⟩I{σr<∞}
+�
+= 1
+ν e−av1
+∞
+�
+n=0
+(n + ν)a−νLn+ν(a
+�
+2λ + |v|2)
+r−νLn+ν(r
+�
+2λ + |v|2)
+�
+Sd−1 er⟨u+v,z⟩Cν
+n(z1)ds(z).
+We can invert the Laplace transform as before and show the following:
+Theorem 5.2. For t > 0 and z ∈ Rd with |z| = r, we have
+Pa(σ(v)
+r
+∈ dt, B(v)(σ(v)
+r ) ∈ dz) = e−av1+⟨v,z⟩e− |v|2
+2 tρ(0)
+a,r(t)dtdsr(z)
++ 2e−av1+⟨v,z⟩e− |v|2
+2 t
+∞
+�
+n=1
+�a
+r
+�nρ(n)
+a,r(t)Tn
+�z1
+r
+�
+dtdsr(z)
+when d = 2 and, when d ≧ 3
+Pa(σ(v)
+r
+∈ dt, B(v)
+σr ∈ dz)
+= 1
+ν e−av1+⟨v,z⟩e− |v|2
+2 t
+∞
+�
+n=0
+�
+n + ν
+��a
+r
+�nρ(n+ν)
+a,r
+(t)Cν
+n
+�z1
+r
+�
+dtdsr(z).
+Next, assuming a > r, we consider the asymptotic behavior of the distribution
+function P(t < σ(v)
+r
+< ∞, B(v)(σ(v)
+r ) ∈ A) as t → ∞ for a fixed A ⊂ Sd−1
+r
+. As is
+easily guessed as earlier, the leading term is given by the first terms of the right
+hand sides in Theorem 5.2.
+Theorem 5.3. For any Borel subset A of Sd−1
+r
+, we have
+Pa(t <σ(v)
+r
+< ∞, B(v)(σ(v)
+r ) ∈ A)
+= 2 log
+�a
+r
+�
+e−av1
+�
+A
+e⟨v,z⟩dsr(z)
+1
+t(log t)2e− |v|2
+2 t(1 + o(1))
+when d = 2 and
+Pa(t <σ(v)
+r
+< ∞, B(v)(σ(v)
+r ) ∈ A)
+= 2L(ν)
+|v|2 e−av1
+�
+A
+e⟨v,z⟩dsr(z)t−ν−1e− |v|2
+2 t(1 + o(1))
+when d ≧ 3, where
+L(ν) =
+r2ν
+2νΓ(ν)
+�
+1 −
+�r
+a
+�2ν�
+.
+9
+
+In order to estimate the higher order terms, we recall from [8] the asymptotic
+result for
+H(ν)(t) :=
+� ∞
+t
+e− |v|2
+2 sρ(ν)
+a,r(s)ds,
+where ρ(ν)
+a,r is the density of the first hitting time τr to r of the Bessel process with
+index ν starting from a: when d = 2,
+H(ν)(t) = 2 log(a/r)
+t(log t)2 e− |v|2
+2 t(1 + o(1))
+and, when d ≧ 3
+H(ν)(t) = 2L(ν)
+|v|2tν+1e− |v|2
+2 t(1 + o(1)).
+(5.1)
+The assertion of Theorem 5.3 follows from the following lemma:
+Lemma 5.4. There exists a constant C, depending on |v| and r, such that
+H(ν)(t) ≦ Cr2ν
+Γ(ν)
+1
+(2t)ν+1e− |v|2
+2 t
+holds for all d ≧ 3.
+Proof. We use (5.1) when d = 3 and d = 4, and assume d ≧ 5 in the following.
+Denote by Py the d-dimensional Wiener measure with starting point y and
+use the same notation σr for the first hitting time to Sd−1
+r
+of the corresponding
+Brownian motion. Moreover, let p(t, x, y) = (2πt)−d/2 exp(−|y − x|2/2t) be the
+Gaussian kernel and set α = |v|2/2 for simplicity. Then we have
+H(ν)(t) = α
+� ∞
+t
+e−αsPa(t < σr ≦ s)ds
+and, setting e = (1, 0, ..., 0),
+Pa(t < σr ≦ s) ≦
+�
+Rd p(t, ae, y)Py(σr ≦ s − t)dy
+≦
+1
+(2πt)d/2
+�
+Rd Py(σr ≦ s − t)dy
+by the Markov property of Brownian motion. Hence we get, after a simple change
+of variables,
+H(ν)(t) ≦
+αe−αt
+(2πt)d/2
+� ∞
+0
+e−αsds
+�
+Rd Py(σr ≦ s)dy.
+Now let Lr be the last hitting time of the Brownian motion to Sd−1
+r
+. Then we
+have
+�
+Rd Py(σr ≦ s) =
+�
+Rd Py(0 < Lr ≦ s)dy +
+�
+Rd Py(σr ≦ s < Lr)dy.
+(5.2)
+10
+
+For the second term of the right hand side, Le Gall [11] has shown
+�
+Rd Py(σr ≦ s < Lr)dy =
+�
+Rd Py(σr ≦ s)Py(σr < ∞)dy,
+(5.3)
+which implies
+�
+Rd Py(σr ≦ s < Lr)dy ≦
+�
+Rd Py(σr < ∞)2dy.
+(5.4)
+This estimate is sufficient for our purpose. We give another elementary proof of
+(5.3) after completing the proof of Lemma 5.4.
+As in the previous section, we denote by µr the equilibrium measure of the
+ball Br. Then we have, for the first term of (5.2),
+� ∞
+0
+e−αsds
+�
+Rd Py(0 < Lr ≦ s)dy =
+� ∞
+0
+e−αsds
+�
+Rd dy
+� s
+0
+dτ
+�
+Rd p(τ, y, z)dµr(z)
+=
+� ∞
+0
+e−αsds
+� s
+0
+dτ
+�
+Rd dµr(z)
+= 2πd/2rd−2
+α2Γ( d
+2 − 1).
+For the second term, we recall
+Py(σr < ∞) = 1 ∧
+� r
+|y|
+�d−2
+.
+Then, by (5.4), we get
+� ∞
+0
+e−αsds
+�
+Rd Py(σr < ∞)2dy = 1
+α
+� �
+|y|≦r
+dy +
+�
+|y|≧r
+� r
+|y|
+�2(d−2)dy
+�
+= 2π
+d
+2rd
+αΓ( d
+2)
+�1
+d +
+1
+d − 4
+�
+.
+Combining the above inequalities, we obtain the assertion of the lemma.
+Proof of (5.3). By the Markov property of Brownian motion, we have
+�
+Rd Py(σr ≦ s < Lr)dy =
+�
+Rd Ey[1{σr≦s}1{Lr>s}]dy
+=
+�
+Rd Ey[1{σr≦s}EBs[1{Lr>0}]]dy
+=
+�
+Rd dy
+�
+Rd Ey[1{σr≦s}PBs(Lr > 0)|Bs = x]p(s, y, x)dx
+=
+�
+Rd dx
+�
+Rd Px(Lr > 0)Py(σr ≦ s|Bs = x)p(s, y, x)dy.
+Note here that Px(Lr > 0) = Px(σr < ∞) and that the time reversal of a pinned
+Brownian motion is again a pinned Brownian motion. Then we obtain
+�
+Rd Py(σr ≦ s < Lr)dy =
+�
+Rd Px(σr < ∞)dx
+�
+Rd Px(σr ≦ s|Bs = y)p(s, x, y)dy
+=
+�
+Rd Px(σr < ∞)Px(σr ≦ s)dx.
+11
+
+Acknowledgment
+The authors were partially supported by JSPS KAKENHI Grant Numbers
+20K03634 and 21K03298.
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+[20] C. Yin and C. Wang, Hitting time and place of Brownian motion with drift,
+The Open Statistics and Probability Journal, 1 2009, 38–42.
+Yuji Hamana
+[email protected]
+Department of Mathematics
+University of Tsukuba
+1-1-1 Tennodai, Tsukuba 305-8571, Japan
+Hiroyuki Matsumoto
+[email protected]
+Department of Mathematics
+Aoyama Gakuin University
+Fuchinobe 5-10-1, Sagamihara 252-5258, Japan
+13
+

69E2T4oBgHgl3EQfPQYR/content/tmp_files/load_file.txt

@@ -0,0 +1,330 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf,len=329
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='03756v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='PR]  10 Jan 2023  Brownian Hitting to Spheres  Yuji Hamana and Hiroyuki Matsumoto  Abstract  Let Sd−1  r  be the sphere in Rd whose center is the origin and the radius  is r, and σr be the first hitting time to it of the standard Brownian motion  {Bt}t≧0, possibly with constant drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' The aim of this article is to show  explicit formulae by means of spherical harmonics for the density of the  joint distribution of (σr, Bσr) and to study the asymptotic behavior of the  distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Introduction and main results  For d ≧ 2, we consier a standard d-dimensional Brownian motion B = {Bt}t≧0  starting from a fixed point (a, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', 0), where we assume a > 0, defined on a  probability space (Ω, F, Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Letting Sd−1  r  be the sphere in Rd with radius r and  centered at the origin, we are concerned with the joint distribution of the first  hitting time σr of B to Sd−1  r  and the hitting place Bσr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  The aim of this article is to show an explicit expression for the density of the  joint distribution by means of the spherical harmonics, that is, the Gegenbauer  and the Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' As an application, we study the asymptotic  behavior of the tail probability Pa(t < σr < ∞, Bσr ∈ A), A ⊂ Sd−1  b  when a > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  The joint density for the Brownian motion with constsnt drift is also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Several authors have studied the joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' It should be first noted  that in the exit problem, that is the case of a < r, the joint density is given  by a solution for a heat equation with the Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' See  Aizenman-Simon [1] for general discussion and Hsu [9] for an explicit expression  in the case of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Wendel [19] has shown a nice result on the expectations  of functions of (σb, Bσb) by using the spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  See Gzyl [4] and  references therein for a recent study on this direction and Uchiyama [17, 18] on  the asymptotic behavior of the distribution functions and its application to the  Wiener sausage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' A similar problem for a Brownian motion with drift has been  discussed in Yin-Wang [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  We proceed to a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Starting from the skew-product representation  of Brownian motion, we use the fact due to Mijatovic-Mramor-Uribe Bravo [13]  that the projections of the Brownian motion on the sphere Sd−1 = Sd−1  1  define  diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We see that, for one-dimensional projections, the eigenvalues  12020 Mathematics Subject Classification: 60J65  keywords : Brownian motion, hitting times and places, spherical harmonics, one-dimensional  diffusion,  1  and the eigenfunctions for the generators are explicitly given by the spherical  harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Combining these facts with the rotation invariance of the probability law of  Brownian motion, we show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' As usual we denote by Iν and Kν the  modified Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We also denote by Cν  n and Tn the Gegenbauer and the  Chebyshev polynomials, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Denote by Ea the expectation with respect to Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then, for λ > 0  and u ∈ Rd, we have  Ea[e−λσre⟨u,Bσr ⟩] =L0(a  √  2λ)  L0(r  √  2λ)  �  S1 er⟨u,z⟩ds(z)  + 2  ∞  �  n=1  Ln(a  √  2λ)  Ln(r  √  2λ)  �  S1 er⟨u,z⟩Tn(z1)ds(z)  when d = 2 and  Ea  �  e−λσre⟨u,Bσr ⟩I{σr<∞}  �  = 1  ν  ∞  �  n=0  (n + ν)a−νLn+ν(a  √  2λ)  r−νLn+ν(r  √  2λ)  �  Sd−1 er⟨u,z⟩Cν  n(z1)ds(z)  when d ≧ 3, where ds is the uniform probability measure on Sd−1, and L = I for  a < r and L = K for a > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Setting u = 0 and noting that the surface integrals of Tn(z1) and Cν  n(z1) vanish  for n ≧ 1, we recover the well known formula for Ea[e−λσr] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  We can invert the joint Laplace transform and obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We denote  by ρ(ν)  a,r(t) the probability density of the first hitting time to r of a Bessel process  with index ν starting from a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For t > 0 and z ∈ Rd with |z| = r, we have  Pa(σr ∈ dt, Bσr ∈ dz) = ρ(0)  a,r(t)dtdsr(z) + 2  ∞  �  n=1  �a  r  �nρ(n)  a,r(t)Tn  �z1  r  �  dtdsr(z)  when d = 2 and  Pa(σr ∈ dt,Bσr ∈ dz)  = 1  ν  ∞  �  n=0  �  n + ν  ��a  r  �nρ(n+ν)  a,r  (t)Cν  n  �z1  r  �  dtdsr(z)  when d ≧ 3, where ν = d−2  2  and dsr is the uniform probability measire on Sd−1  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  The authors [8] have shown another expression for the joint Laplace transform,  from which we can prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' In the next Section 2 we study  the first coordinate or the one-dimensional projection of the Brownian motion  on Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  We give proofs of the theorems mentioned above in Section 3 and,  the asymptotic behavior of Pa(t < σr < ∞, Bσr ∈ A), A ⊂ Sd−1  r  , as t → ∞  is investigated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' In the final Section 5, we deal with the Brownian  motion with constant drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Projection of Brownian motion on sphere  Let θ = {θ(t)}t≧0 be a Brownian motion on Sd−1, which corresponds to the  Laplace-Beltrami operator on Sd−1, endowed with the usual Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Mijatovic-Mramor-Uribe Bravo [13] has shown that the projections of θ are dif-  fusion processes which are realized as unique solutions of stochastic differential  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' This fact, especially on the one-dimensional projections, is fundamen-  tal in our argument and we recall the result in this special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' The first coordinate {θ1(t)}t≧0 of θ is a diffusion process on  (−1, 1) whose generator is  Gd = 1  2(1 − x2) d2  dx2 − d − 1  2  x d  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  We see easily that the boundaries ±1 are regular and reflecting when d = 2  and they are entrance ones when d ≧ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' The eigenvalues and the eigenfunctions of  Gd are explicitly given and we have the eigenfunction expansion for the transition  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Since these play important roles in the following sections, we now  recall some fundamental facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For details of the Chebyshev and the Gegenbauer  polynomials below, we refer to [3, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Write  Gd =  1  2(1 − x2)  d−3  2  d  dx  �  1  (1 − x2)− d−1  2  d  dx  �  and let dm(x) = 2(1 − x2)  d−3  2 dx be the canonical (speed) measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Note that m  is a finite measure on (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Moreover, we take  s(x) =  � x  0  (1 − y2)− d−1  2 dy  as the scale function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  When d = 2, s(±1) are both finite and the boundaries are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  The  Chebyshev polynomial Tn(x) = cos(n arccos x) satisfies  G2Tn = −n2  2 Tn  and  d  dsTn(±1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Moreover the orthogonality relation is given by  � 1  −1  Tm(x)Tn(x)  dx  √  1 − x2 =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  m ̸= n  π  2  m = n ̸= 0  π  m = n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Hence, setting  φ0  0(x) =  1  √  2π,  φ0  n(x) =  1  √πTn(x)  (n ≧ 1),  3  we see that {φ0  n}∞  n=0 gives rise to an orthonormal basis of L2(dm) and that the  transition density p2(t, x, y) of {θ1(t)} with respect to dm is given by  p2(t, x, y) = 1  2π + 1  π  ∞  �  n=1  e− 1  2n2tTn(x)Tn(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1)  For d ≧ 3, the eigenfunctions are given by the Gegenbauer polynomials Cν  n  defined by  ∞  �  n=0  snCν  n(x) =  1  (1 + s2 − 2sx)ν ,  |s| < 1,  where ν = (d − 2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' In fact, we have  GdCν  n = −1  2n(n + 2ν)Cν  n  and  d  dsCν  n(±1) = 0  and the orthogonality relation  � 1  −1  Cν  m(x)Cν  n(x)(1 − x2)ν− 1  2dx = δm,n  πΓ(n + 2ν)  22ν−1(n + ν)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' (Γ(ν))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Hence, setting  φν  n(x) =  � (n + ν)n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  πΓ(n + 2ν)  � 1  22ν−1Γ(ν)Cν  n(x),  we obtain an orthonormal basis {φν  n}∞  n=0 of L2(dm) and an eigenfunction expsn-  sion for the transition density pd(t, x, y) of {θ1(t)} with respect to dm,  pd(t, x, y) =  ∞  �  n=0  e− 1  2n(n+2ν)tφν  n(x)φν  n(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2  We use the same notation as those in Section 1 and start the argument from  the skew-product representation of the standard Brownian motion B = {Bt}t≧0:  there exists a d-dimensional Bessel process R = {Rt}t≧0 (with index ν = (d−2)/2)  and a Brownian motion θ = {θ(t)}t≧0 on Sd−1, independent of R, such that  Bt = Rtθ(Ξt),  Ξt =  � t  0  ds  R2s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  B0 = (a, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', 0) means R0 = a and θ(0) = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' By the independence of  R and θ, we have  Ea[e−λσre⟨u,Bσr ⟩] = E(ν)  a [e−λτrEa[er⟨u,θ(t)⟩]  ���  t=Ξτr  ],  where E(ν)  a [ · ] denotes the expectation with respect to the probability law of R  and τr is the first hitting time of R to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  4  First we prove the theorems when d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Writing θ(t) = (θ1(t), θ2(t)) and  u = (u1, u2), we have by the rotation invariance of the law of standard Brownian  motion  Ea[er⟨u,θ(t)⟩] = Ea[eru1θ1(t)Ea[eru2θ2(t)|θ1(t)]]  =  � 1  −1  eru1y 1  2  �  eru2√  1−y2 + e−ru2√  1−y2�  P(θ1(t) ∈ dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Hence formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1) implies  Ea[er⟨u,θ(t)⟩]  = 1  2π  � 1  −1  eru1y 1  2  �  eru2√  1−y2 + e−ru2√  1−y2�  2dy  �  1 − y2  + 1  π  ∞  �  n=1  e− 1  2n2t  � 1  −1  eru1y 1  2  �  eru2√  1−y2 + e−ru2√  1−y2�  Tn(y)  2dy  �  1 − y2  since Tn(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' The change of order of the intengal and the sum is easily justified  because |Tn(y)| ≦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We can write the integrals on the right hand side as surface  integrals and obtain  Ea[er⟨u,θ(t)⟩] =  �  S1 er⟨u,z⟩ds(z) + 2  ∞  �  n=1  e− 1  2n2t  �  S1 er⟨u,z⟩Tn(z1)ds(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Now, recalling the formula ([2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='407])  E(0)  a [e−λτr− 1  2 n2Ξτr] = Ln(a  √  2λ)  Ln(r  √  2λ)  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1)  we obtain the assertion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1 when d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Next note another formula ([2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='398])  E(µ)  a [e−λτr] =  � ∞  0  e−λtρ(µ)  a,r(t)dt = a−µLµ(a  √  2λ)  r−µLµ(r  √  2λ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Then we obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2 when d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Again we can easily show the absolute  convergence and justify the change of the sum and the integrals in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Next we prove the theorems in the case of d ≧ 3, when, for the spherical  Brownian motion θ, the conditional distribution of (θ2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', θd(t)) given θ1(t) =  ξ1 is the uniform distribution on the sphere Sd−2  √  1−ξ2  1 with raduis  �  1 − ξ2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Hence,  writing u = (u1, u′), θ = (θ1, θ′) ∈ R × Rd−1, we have  Ea[e⟨u,rθ(t)⟩] = Ea  �  eru1θ1(t)  �  Sd−2 er√  1−θ1(t)2⟨u′,ξ′⟩ ds(ξ′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  5  By using the facts on the Gegenbauer polynomials given in the previous section  and writing the double integral as a surface integral, we obtain, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2)  Ea[e⟨u,rθ(t)⟩]  =  ∞  �  n=0  e− 1  2n(n+2ν)tφν  n(1)  � 1  −1  φν  n(ξ1)eru1ξ12(1 − ξ2  1)  d−3  2 dξ1  ×  �  Sd−2 er√  1−ξ2  1⟨u′,ξ′⟩ vol(dξ′)  vol(Sd−2)  =  ∞  �  n=0  (n + ν)22ν−1Γ(ν)2 vol(Sd−1)  πΓ(2ν) vol(Sd−2)  e− 1  2 n(n+2ν)t  �  Sd−1 Cν  n(w1)er⟨u,w⟩ds(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  We have used the formula Cν  n(1) =  �2ν+n−1  n  �  = Γ(n + 2ν)/(n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='Γ(2ν)), and also the  estimate  max  |y|≦1 |Cν  n(y)| = Cν  n(1) ≦ Cn2ν−1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2)  for some constant C (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', [12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='218, 225]) to justify the change of the order  of the sum and the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Moreover, recalling the foumulae  vol(Sd−1) = 2π  d  2  Γ( d  2)  and  Γ(2ν) = 22ν  2√πΓ(ν)Γ(ν + 1  2),  we obtain  Ea[e⟨u,rθ(t)⟩] = 1  ν  ∞  �  n=0  (n + ν)e− 1  2n(n+2ν)t  �  Sd−1 Cν  n(w1)er⟨u,w⟩ds(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Now, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1), we obtain the assertion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2 is proven  in the same way as in the case of d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Asymptotic behavior of distribution function  In this section, assuming a > r and applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2, we study the asymp-  totic behavior of the distribution function Pa(t < σr < ∞, Bσr ∈ A) as t → ∞  for a fixed Borel subset A of the sphere Sd−1  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We use the same notation as in the  previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  In a course of study on the first hitting times of Bessel processes, the authors  [6, 7] have shown the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Consider a Bessel process with index ν and  starting from a defined on some probability space (Ω′, F ′, Q(ν)  a ) and let τr be its  hitting time to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then the asymptotic behavior of Q(ν)  a (t < τr < ∞) when a > r  is given by  Q(0)  a (t < τr < ∞) = 2 log(a/r)  log t  (1 + o(1))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1)  when d = 2 and  Q(ν)  a (t < τr < ∞) = κ(ν)t−ν(1 + o(1)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2)  6  when d ≧ 3, where the constant κ(ν) is given by  κ(ν) =  1  Γ(ν + 1)  � r3  2a  �ν��a  r  �ν  −  �a  r  �−ν�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Applying these results with some estimates for the remainder terms, we show  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For any Borel subset A of Sd−1  r  ,  Pa(t < σr < ∞, Bσr ∈ A) = 2 log(a/r)  log t  sr(A)(1 + o(1))  holds as t → ∞ when d = 2 and  Pa(t < σr < ∞, Bσr ∈ A) = κ(ν)sr(A)t−ν(1 + o(1))  holds when d ≧ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For the distribition function Q(ν)  a (t < τr < ∞) of the first hitting  time of the Bessel process, Hamana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' [5] has shown a precise asymptotic  expansion and, using the results, we can show asymptotic expansions for our  joint distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' The details will be published elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  For a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1, we show the following estimate for the tail prob-  ability of σr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Assume d ≧ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then, for t > 0, we have  Pa(t < σr < ∞) ≦  r2ν  2νΓ(ν + 1)tν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Let Lr be the last hitting time of the Brownian motion B to the spehere  Sd−1  r  :  Lr = sup{s > 0 : |Bs| = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  As usual we set Lr = 0 when B does not hit Sd−1  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then we have  Pa(t < σr < ∞) ≦ Pa(t < Lr < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Denote by µr the equilibrium measure of the ball Br with radius r and centered  at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then it is well known ([16]) that  Pa(t < Lr < ∞) =  � ∞  t  ds  �  Rd  1  (2πs)d/2e− |x−a|2  2s dµr(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Recalling now that the capacity of Br is µr(Rd) = 2π  d  2rd−2/Γ( d  2 − 1), we see  Pa(t < Lr < ∞) ≦  � ∞  t  ds  �  Rd  1  (2πs)d/2dµr(x) =  r2ν  2νΓ(ν + 1)tν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  7  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For transient one-dimensional diffusion processes, the densities of  the last hitting times are written by means of the transition densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' This is  the case of the Bessel processes with dimensions d ≧ 3 and, moreover, we have  explicit expressions for the transition densities We can give another proof for  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3 by using these facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  We can now give a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Note that the infinite sum below  for the expression for the joint distribution is absoletely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  For d = 2, we have by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2  Pa(t < σr < ∞, Bσr ∈ A) = Q(0)  a (τr > t)sb(A) + It,  where  It = 2  ∞  �  n=1  �a  r  �nQ(n)  a (t < τr < ∞)  �  A  Tn(z1  r )dsr(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Assume t > 1 and note |Tn(x)| = | cos(n arccos x)| ≦ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3, we  get  |It| ≦ 2  ∞  �  n=1  �a  r  �n  r2n  2nΓ(n + 1)tn ≦ 2  t  ∞  �  n=1  �ar  2  �n 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' ≦ 2  t e  ar  2  and, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1), the assertion of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  For d ≧ 3, we have  Pa(t < σr < ∞, Bσr ∈ A) = Q(ν)  a (t < τr < ∞)sr(A) + Jt,  where  Jt = 1  ν  ∞  �  n=1  (n + ν)  �a  r  �nQ(n+ν)  a  (t < τr < ∞)  �  A  Cν  n(z1  r )dsr(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Hence, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2) with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2), we see Jt = O(t−1−ν) and the  assertion of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Brownian motion with drift  Let B = {Bt}t≧0 be a standard d-dimensional Brownian motion starting from  x = (a, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', 0) as before and, for a constant vector v ∈ Rd, B(v) = {B(v)(t)}t≧0  be a Brownian motion with drift v defined by B(v)(t) = Bt + tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We denote by  σ(v)  r  the first hitting time of B(v) to the sphere Sd−1  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  The Cameron-Martin theorem and the strong Markov property of Brownian  motion imply  Ea  �  e−λσ(v)  r e⟨u,B(v)(σ(v)  r  )⟩I{σ(v)  r  <∞}  �  = e−av1Ea  �  e−(λ+ |v|2  2 )σre⟨u+v,Bσr ⟩I{σr<∞}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Hence we can apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1 to the right hand side and obtain the following:  8  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For λ > 0 and u ∈ Rd, we have  Ea  �  e−λσ(v)  r e⟨u,B(v)(σ(v)  r  )⟩I{σr<∞}  �  = e−av1  �  L0(a  �  2λ + |v|2)  L0(r  �  2λ + |v|2)  �  S1 er⟨u+v,z⟩ds(z)  + 2  ∞  �  n=1  Ln(a  �  2λ + |v|2)  Ln(r  �  2λ + |v|2)  �  S1 er⟨u+v,z⟩Tn(z1)ds(z)  �  when d = 2 and, when d ≧ 3,  Ea  �  e−λσ(v)  r e⟨u,B(v)(σ(v)  r  )⟩I{σr<∞}  �  = 1  ν e−av1  ∞  �  n=0  (n + ν)a−νLn+ν(a  �  2λ + |v|2)  r−νLn+ν(r  �  2λ + |v|2)  �  Sd−1 er⟨u+v,z⟩Cν  n(z1)ds(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  We can invert the Laplace transform as before and show the following:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For t > 0 and z ∈ Rd with |z| = r, we have  Pa(σ(v)  r  ∈ dt, B(v)(σ(v)  r ) ∈ dz) = e−av1+⟨v,z⟩e− |v|2  2 tρ(0)  a,r(t)dtdsr(z)  + 2e−av1+⟨v,z⟩e− |v|2  2 t  ∞  �  n=1  �a  r  �nρ(n)  a,r(t)Tn  �z1  r  �  dtdsr(z)  when d = 2 and, when d ≧ 3  Pa(σ(v)  r  ∈ dt, B(v)  σr ∈ dz)  = 1  ν e−av1+⟨v,z⟩e− |v|2  2 t  ∞  �  n=0  �  n + ν  ��a  r  �nρ(n+ν)  a,r  (t)Cν  n  �z1  r  �  dtdsr(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Next, assuming a > r, we consider the asymptotic behavior of the distribution  function P(t < σ(v)  r  < ∞, B(v)(σ(v)  r ) ∈ A) as t → ∞ for a fixed A ⊂ Sd−1  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' As is  easily guessed as earlier, the leading term is given by the first terms of the right  hand sides in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' For any Borel subset A of Sd−1  r  , we have  Pa(t <σ(v)  r  < ∞, B(v)(σ(v)  r ) ∈ A)  = 2 log  �a  r  �  e−av1  �  A  e⟨v,z⟩dsr(z)  1  t(log t)2e− |v|2  2 t(1 + o(1))  when d = 2 and  Pa(t <σ(v)  r  < ∞, B(v)(σ(v)  r ) ∈ A)  = 2L(ν)  |v|2 e−av1  �  A  e⟨v,z⟩dsr(z)t−ν−1e− |v|2  2 t(1 + o(1))  when d ≧ 3, where  L(ν) =  r2ν  2νΓ(ν)  �  1 −  �r  a  �2ν�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  9  In order to estimate the higher order terms, we recall from [8] the asymptotic  result for  H(ν)(t) :=  � ∞  t  e− |v|2  2 sρ(ν)  a,r(s)ds,  where ρ(ν)  a,r is the density of the first hitting time τr to r of the Bessel process with  index ν starting from a: when d = 2,  H(ν)(t) = 2 log(a/r)  t(log t)2 e− |v|2  2 t(1 + o(1))  and, when d ≧ 3  H(ν)(t) = 2L(ν)  |v|2tν+1e− |v|2  2 t(1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1)  The assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3 follows from the following lemma:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' There exists a constant C, depending on |v| and r, such that  H(ν)(t) ≦ Cr2ν  Γ(ν)  1  (2t)ν+1e− |v|2  2 t  holds for all d ≧ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='1) when d = 3 and d = 4, and assume d ≧ 5 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Denote by Py the d-dimensional Wiener measure with starting point y and  use the same notation σr for the first hitting time to Sd−1  r  of the corresponding  Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Moreover, let p(t, x, y) = (2πt)−d/2 exp(−|y − x|2/2t) be the  Gaussian kernel and set α = |v|2/2 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then we have  H(ν)(t) = α  � ∞  t  e−αsPa(t < σr ≦ s)ds  and, setting e = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', 0),  Pa(t < σr ≦ s) ≦  �  Rd p(t, ae, y)Py(σr ≦ s − t)dy  ≦  1  (2πt)d/2  �  Rd Py(σr ≦ s − t)dy  by the Markov property of Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Hence we get, after a simple change  of variables,  H(ν)(t) ≦  αe−αt  (2πt)d/2  � ∞  0  e−αsds  �  Rd Py(σr ≦ s)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Now let Lr be the last hitting time of the Brownian motion to Sd−1  r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then we  have  �  Rd Py(σr ≦ s) =  �  Rd Py(0 < Lr ≦ s)dy +  �  Rd Py(σr ≦ s < Lr)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2)  10  For the second term of the right hand side, Le Gall [11] has shown  �  Rd Py(σr ≦ s < Lr)dy =  �  Rd Py(σr ≦ s)Py(σr < ∞)dy,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3)  which implies  �  Rd Py(σr ≦ s < Lr)dy ≦  �  Rd Py(σr < ∞)2dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='4)  This estimate is sufficient for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' We give another elementary proof of  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3) after completing the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  As in the previous section, we denote by µr the equilibrium measure of the  ball Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then we have, for the first term of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='2),  � ∞  0  e−αsds  �  Rd Py(0 < Lr ≦ s)dy =  � ∞  0  e−αsds  �  Rd dy  � s  0  dτ  �  Rd p(τ, y, z)dµr(z)  =  � ∞  0  e−αsds  � s  0  dτ  �  Rd dµr(z)  = 2πd/2rd−2  α2Γ( d  2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  For the second term, we recall  Py(σr < ∞) = 1 ∧  � r  |y|  �d−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Then, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='4), we get  � ∞  0  e−αsds  �  Rd Py(σr < ∞)2dy = 1  α  � �  |y|≦r  dy +  �  |y|≧r  � r  |y|  �2(d−2)dy  �  = 2π  d  2rd  αΓ( d  2)  �1  d +  1  d − 4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Combining the above inequalities, we obtain the assertion of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' By the Markov property of Brownian motion, we have  �  Rd Py(σr ≦ s < Lr)dy =  �  Rd Ey[1{σr≦s}1{Lr>s}]dy  =  �  Rd Ey[1{σr≦s}EBs[1{Lr>0}]]dy  =  �  Rd dy  �  Rd Ey[1{σr≦s}PBs(Lr > 0)|Bs = x]p(s, y, x)dx  =  �  Rd dx  �  Rd Px(Lr > 0)Py(σr ≦ s|Bs = x)p(s, y, x)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  Note here that Px(Lr > 0) = Px(σr < ∞) and that the time reversal of a pinned  Brownian motion is again a pinned Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Then we obtain  �  Rd Py(σr ≦ s < Lr)dy =  �  Rd Px(σr < ∞)dx  �  Rd Px(σr ≦ s|Bs = y)p(s, x, y)dy  =  �  Rd Px(σr < ∞)Px(σr ≦ s)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  11  Acknowledgment  The authors were partially supported by JSPS KAKENHI Grant Numbers  20K03634 and 21K03298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
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+page_content=', Academic Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
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+page_content=' Hamana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Kaikura and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Shinozaki, Asymptotic expansions for the  first hitting times of Bessel processes, Opuscula Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=', 41 (2021), 509–537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content='  [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
+page_content=' Hamana and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E2T4oBgHgl3EQfPQYR/content/2301.03756v1.pdf'}
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+arXiv:2301.00540v1  [math.CA]  2 Jan 2023
+Coefficient characterization of linear differential equations
+with maximal symmetries
+J.C. Ndogmo
+School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050,
+South Africa
+Abstract
+A characterization of the general linear equation in standard form admit-
+ting a maximal symmetry algebra is obtained in terms of a simple set of
+conditions relating the coefficients of the equation. As a consequence, it is
+shown that in its general form such an equation can be expressed in terms of
+only two arbitrary functions, and its connection with the Laguerre-Forsyth
+form is clarified. The characterizing conditions are also used to derive an
+infinite family of semi-invariants, each corresponding to an arbitrary order
+of the linear equation. Finally a simplifying ansatz is established, which
+allows an easier determination of the infinitesimal generators of the induced
+pseudo group of equivalence transformations, for all the three most general
+canonical forms of the equation.
+Keywords:
+Coefficient characterization, maximal symmetry algebra,
+canonical form, induced equivalence group, infinitesimal generators
+2010 MSC: 70G65, 34C20
+1. Introduction
+By a result of Lie [1], a linear ordinary differential equation (ode) of a
+general order n is known to have a symmetry algebra of maximal dimension
+dn if it is reducible by a point transformation to the equation y(n) = 0, which
+will henceforth be referred to as the canonical form of the linear equation.
+In a much recent paper Krause and Michel [2] proved the converse of this
+result and also showed that a linear equation is iterative if and only if its
+symmetry algebra has the maximal dimension dn. (By the cited result of Lie
+Email address: [email protected] (J.C. Ndogmo)
+
+[1], dn = n+4 for n ≥ 3). Characterizing linear equations having a symmetry
+algebra of maximal dimension is therefore the same as characterizing linear
+equations that are reducible by a point transformation to the canonical form.
+The latter characterization for the third-order equation y(3) + c2 y′′ + c1 y′ +
+c0 y = 0 is due to Lie [3] and Laguerre [4] who showed independently that
+this equation is reducible to the canonical form if and only if its coefficients
+satisfy the equation
+54c0 − 18c1c2 + 4c3
+2 − 27c′
+1 + 18c2c′
+2 + 9c′′
+2 = 0.
+(1)
+This characterization also clearly applies to all nonlinear odes which are
+linearizable by point transformations [5, 6], as the latter transformations do
+not alter the dimension of the symmetry algebra.
+In this paper, we extend this characterization to equations of higher or-
+ders. It turns out that for each equation of order n there will be n − 2
+characterizing equations, and the limitation of our presentation of the char-
+acterizing equations only up to the order five is simpy due to their very
+large size. However, we give a description of the method for deriving this
+characterization for equations of any order. The derivation of these char-
+acterizing equations is also based on the canonical normal form of linear
+equations admitting a maximal symmetry algebra that was obtained in [5]
+from a symmetry approach, and in [7] from an iterative approach. These
+characterizing equations therefore also represent a generalization of the re-
+sults of [5] and [7]. We then deduce that the most general form of a linear
+equation admitting a maximal symmetry algebra can be expressed in stan-
+dard form in terms of only two arbitrary functions. We also deduce that the
+Laguerre-Forsyth form of a linear equation reduces to the canonical form if
+and only if the equation has maximal symmetries.
+Although we do not give the characterizing equations for each linear
+equation of order n, we note however that among the n − 2 characterizing
+equations exactly one of them represents a semi-invariant of the equation,
+that is a function of the coefficients of the equation whose expression does
+not change when the dependent variable is transformed.
+We obtain an
+expression for these semi-invariants for equations of all orders and describe
+some of their properties.
+Finally, using some simplifying assumptions and the method of [8], we
+give expressions for both the symmetry generator Xn of GS and X0
+n of the
+induced pseudo group of transformations Gc, and for all three most general
+canonical forms of linear equations of a general order n. Here, GS denotes
+the symmetry group of the general linear equation in which the arbitrary
+2
+
+functions are considered as additional dependent variables.
+2. Coefficient characterization
+A method based on a symmetry approach has been proposed in [5] for
+characterizing the coefficients of linear ordinary differential equations (odes)
+that admit a maximal symmetry algebra, but only for equations in reduced
+normal form (in which the term of second highest order vanishes). In a more
+recent paper [7] a similar characterization based on an iterative approach was
+proposed, in which according to a result of Krause and Michel [2] a linear
+equation admitting a maximal symmetry is simply viewed as an iterative
+equation. By iterative equation, we mean an equation of the form
+Ψn[y] = 0,
+y = y(x),
+n ≥ 1
+(2a)
+where
+Ψ1[y] = ry′ + sy,
+Ψn[y] = Ψn−1 [Ψ[y]] ,
+(2b)
+and where r = r(x) and s = s(x) are the parameters of the source equation
+Ψ1[y] = 0. This characterization shows that in its reduced normal form, a
+general linear equation depends solely on one arbitrary function a = a(x).
+For equations of orders three to five, the corresponding equations are given
+as follows:
+y(3) + ay′ + a′
+2 y = 0
+(3a)
+y(4) + ay′′ + a′y′ +
+� 3
+10a′′ +
+9
+100a2
+�
+y = 0
+(3b)
+y(5) + ay(3) + 3
+2a′y′′ +
+� 9
+10a′′ + 16
+100a2
+�
+y′ +
+�1
+5a(3) + 16
+100aa′
+�
+y = 0.
+(3c)
+However, as a linear equation need not occur in its reduced normal form,
+but rather in the most general standard form, it is thus useful to obtain the
+corresponding characterization for equations in standard form. We let the
+general linear equation be given in standard form as
+∆(x, y(n); C) ≡ y(n) + cn−1 y(n−1) + cn−2 y(n−2) + · · · + c0 y = 0.
+(4)
+3
+
+where C = (c0, . . . , cn−1). Suppose that such an equation has a symmetry
+algebra of maximal dimension and let its corresponding reduced normal form
+be given by
+y(n) + Bn−2 y(n−2) + Bn−3 y(n−3) + · · · + B0 y = 0,
+(5)
+where the Bj for j = 0, . . . , n−2 are its coefficients and depend as already
+noted above on a single arbitrary function a = Bn−2 and its derivatives. Let
+y(n) + An−1 y(n−1) + An−2 y(n−2) + · · · + A0 y = 0
+(6)
+be the corresponding standard form of (5), which may be obtained by a
+transformation of the form
+y �→ ye− 1
+n
+� x
+x0 An−1dx.
+(7)
+Then (4) and (6) must be identical, and in particular the nonzero coef-
+ficient An−1 introduced by the transformation (7) satisfies An−1 = cn−1,
+and more generally we have
+cj = Aj,
+for j = 0, . . . , n − 1.
+(8)
+Note that the coefficients cj in (4) are mere symbols and we wish to find a
+relationship among them. Given that in (5) the function Bn−2 is precisely
+the arbitrary function a(x) labeling the equation, it can be shown by a
+recursive procedure, or even by induction on n that
+An−2 = a + n − 1
+2n
+c2
+n−1 + n − 2
+2
+c′
+n−1.
+Therefore, solving the equation cn−2 = An−2 for a gives
+a = cn−2 −
+�n − 1
+2n
+c2
+n−1 + n − 2
+2
+c′
+n−1
+�
+.
+(9)
+Consequently, the characterizing equations for linear equations in standard
+form with maximal symmetry algebra are given by the remaining n − 2
+equations
+cj = Aj,
+j = 0, . . . , n − 3,
+(10)
+in which the function a and its derivatives are substituted with the corre-
+sponding expressions given by (9).
+Proposition 1. If a linear equation in standard form (4) has maximal sym-
+metry, then in its general form it may be expressed in terms of only two ar-
+bitrary functions, namely the functions cn−1 and cn−2, and their derivatives.
+4
+
+Proof. The result readily follows from the fact that the functions Aj in (10)
+then depend only on a and its derivatives, while (9) shows that the function
+a depends precisely on cn−1, cn−2, and their derivatives.
+Corollary 1. A linear equation in standard form (4) with cn−1 = cn−2 = 0
+has maximal symmetry algebra if and only if cj = 0 for all j.
+In other
+words a linear equation has maximal symmetry algebra if and only if its
+Laguerre-Forsyth form corresponds to the canonical equation y(n) = 0.
+Proof. After all a Laguerre transformation is also a point transformation
+although it cannot always be explicitly constructed for a given equation.
+Since equations equivalent under point transformation have similar Lie al-
+gebras, it readily follows that if the Laguerre-Forsyth form of an equation is
+y(n) = 0, then the equation has maximal symmetry algebra. The converse
+of the corollary is a direct application of proposition 1, and the fact that in
+(10) the cj turn out to be polynomial functions with no constant terms of
+cn−1, cn−2, and their derivatives.
+As an immediate consequence of the corollary, linear equations such as
+y(3) +f(x)y = 0 or y(4)+f(x)y′ = 0 have maximal symmetry algebras if and
+only if the function f(x) vanishes identically. We now make use of (10) and
+(9) to explicitly derive the characterizing equations for maximal symmetry
+algebras for equations of orders three to five.
+For n = 3, it is readily found that in (6) we have
+A0 = 1
+54
+�
+18ac2 + 2c3
+2 + 27a′ + 18c2c′
+2 + 18c′′
+2
+�
+,
+(11)
+while the corresponding expression for a in (9) reduces to
+a = c1 −
+�c2
+2
+3 + c′
+2
+2
+�
+.
+(12)
+Applying (12) into (11) and substituting the resulting expression for A0
+into (10) gives exactly the already cited equation (1) found by Lie [3] and
+Laguerre [4] and given by
+54c0 − 18c1c2 + 4c3
+2 − 27c′
+1 + 18c2c′
+2 + 9c′′
+2 = 0.
+The most general form of a linear third-order equation admitting a maximal
+symmetry algebra can thus be expressed in terms of only two arbitrary
+functions c1(x) and c2(x) in the form of
+y(3) + c2 y′′ + c1 y′ + 1
+54
+�
+18c1c2 − 4c3
+2 + 27c′
+1 − 18c2c′
+2 − 9c′′
+2
+�
+y = 0.
+(13)
+5
+
+Equation (13) naturally reduces to (3a) for c2 = 0.
+For n = 4, we successively get
+a = 1
+8
+�
+8c2 − 3c2
+3 − 12c′
+3
+�
+(14a)
+A1 = 1
+2
+�
+ac3 + c3
+3
+16 + a′ + 3
+4c3c′
+3 + c′′
+3
+�
+(14b)
+6400A0 = 576a2 + 400a(c2
+3 + 4c′
+3)
++ 5
+�
+5c4
+3 + 120c2
+3c′
+3 + 320c3(a′ + c′′
+3)
+�
++ 80
+�
+15c′2
+3 + 24a′′ + 20c(3)
+3
+�
+.
+(14c)
+Substituting (14a) into (14b) and (14c) gives the two equations
+8c1+ = 4c2c3 − c3
+3 + 8c′
+2 − 6c3c′
+3 − 4c′′
+3
+(15a)
+1600c0 = 144c2
+2 − 11c4
+3 + 400c3c′
+2 − 288c2
+3c′
+3 − 336c′2
+3
+− 8c2(c2
+3 + 4c′
+3) + 480c′′
+2 − 560c3c′′
+3 − 320c(3)
+3
+(15b)
+which represent the characterizing equations for maximal symmetry algebra
+for equations of order 4. Note that conversely any linear fourth order equa-
+tion whose coefficients satisfy (15) must be iterative, which is why conditions
+such as (15) are termed characterizing equations. Indeed, if the coefficients
+of a fourth order equation of the form (4) satisfy (5), then its reduced nor-
+mal form has, after the substitution of the expressions for c0 and c1 given
+by (5) in terms of c2, c3, and their derivatives, the form
+w(4) + Q2w′′ + Q1w′ + Q0w = 0
+(16a)
+where
+Q2 = c2 − 3
+8(c2
+3 + 4c′
+3)
+(16b)
+Q1 = c′
+2 − 3
+4(c3c′
+3 + 2c′′
+3)
+(16c)
+Q0 =
+3
+6400(192c2
+2 + 27c4
+3 − 48c′2
+3 − 144c2(c2
+3 + 4c′
+3))
++
+3
+6400(27c4
+3 + 216c2
+3c′
+3 + 640c′′
+2 − 480c3c′′
+3 − 960c′′′
+3 ).
+(16d)
+The coefficients Qj thus obtained clearly satisfy the conditions
+Q1 = Q′
+2
+and
+Q0 = ( 3
+10Q′′
+2 +
+9
+100Q2
+2)
+6
+
+prescribed by (3b) for iterative equations, as required.
+For equations of order n = 5, by proceeding as above for the orders three
+and four, we obtain the following n − 2 = 3 characterizing equations
+c2 = (30c3c4 − 8c3
+4 + 75c′
+3 − 60c4c′
+4 + 50c′′
+4)/50
+(17a)
+1250 c1 = +200c2
+3 − 18c4
+4 + 750c4c′
+3 − 580c2
+4c′
+4 − 850c′2
+4
+− 10c3(c2
+4 + 5c′
+4) + 1125c′′
+3 − 1400c4c′′
+4 − 1000c(3)
+4
+(17b)
+6250 c0 = 200c2
+3c4 + 14c5
+4 − 25c2
+4c′
+3 + 40c3
+4c′
+4
+− 125c′
+3c′
+4 − 750c4c′2
+4 + 1125c4c′′
+3 − 850c2
+4c′′
+4
+− 2750c′
+4c′′
+4 + 1250c(3)
+3
+− 2000c4c(3)
+4
+− 1250c(4)
+4
+− 10c3(11c3
+4 + 100c′
+3 − 85c4c′
+4 − 75c′′
+4).
+(17c)
+3. Semi-invariants of linear equations
+The group of equivalence transformations of the general linear equation
+(4) is given by invertible point transformations of the form
+x = f(z),
+y = g(z)w(z),
+(18)
+and they preserve the linearity and the homogeneity of the equation. Let
+w(n) + Qn−1 w(n−1) + Qn−2 w(n−2) + · · · + Q0 w = 0
+(19)
+be the transformed version of (4) under (18). By a semi-invariant of (4)
+we shall mean a function F = F(c0, c1, . . . , cn−1) of the coefficients of the
+equation which have the same expression for the transformed equation when
+the dependent variable (alone) changes. It is well known that under (18)
+the expression of the semi-invariant for the transformed equation is related
+to that for the original equation [9, 10] by the equality
+F(Q0, Q1, . . . , Qn−1) =
+�dx
+dz
+�µ
+F(c0, c1, . . . , cn−1),
+(20)
+where µ is an integer.
+In this case we say that the semi-variant F has
+index µ. To each expression of the form dkcj/dxk, let us assign the weight
+(n − j) + k, and we let this weight function be multiplicative so that the
+product cpcq has weight (n − p) + (n − q). It is well known that for a given
+semi-invariant all terms have the same weight and that this weight coincides
+with the index of the semi-invariant [9, 10] .
+7
+
+A closer look at the set of characterizing equations (10) shows that pre-
+cisely one of them corresponds to a semi-invariant of the equation, namely
+the relation cn−3 = An−3, which gives rise to the semi-invariant F =
+An−3 − cn−3.
+First of all, using the method of either [7] or [5], it can be proved that
+the coefficient Bn−3 in (5) satisfies Bn−3 = n−2
+2 a′. Consequently, using the
+expression of the function a in (9) it follows by induction on n that the
+coefficient An−3 in (6) is given by
+An−3 =n − 2
+n
+cn−1cn−2 − (n − 1)(n − 2)
+3n2
+c3
+n−1 + n − 2
+2
+c′
+n−2
+− (n − 1)(n − 2)
+2n
+cn−1c′
+n−1 − (n − 1)(n − 2)
+12
+c′′
+n−1,
+(21)
+so that the corresponding invariant function In has expression
+In =n − 2
+n
+cn−1cn−2 − (n − 1)(n − 2)
+3n2
+c3
+n−1 + n − 2
+2
+c′
+n−2
+− (n − 1)(n − 2)
+2n
+cn−1c′
+n−1 − (n − 1)(n − 2)
+12
+c′′
+n−1 − cn−3.
+(22)
+The fact that the function In = In(c0, c1, . . . , cn−1) in (22) is a semi-invariant
+can readily be verified. First each term in this expression has weight three,
+and we readily see that
+In(Q0, Q1, . . . , Qn−1) = f ′(z)3In(c0, c1, . . . , cn−1),
+which proves the assertion.
+Although the invariant functions In in (22) are originally defined only for
+n ≥ 3, their expression shows that they vanish identically for n = 1 or n = 2,
+by letting cj = 0 for j < 0. This vanishing can be interpreted by the fact that
+all first order and all second order linear equations are all equivalent through
+a point transformation to the equations y′ = 0 and y′′ = 0, respectively, and
+therefore they do not have nontrivial invariant functions.
+On the other hand it should be noted that the other equations in the
+characterizing system (10) do not give rise to invariant functions except for
+the value j = n − 3 in that system of equations. Indeed, denote collectively
+by C and Q the coefficients in equations (4) and (19), respectively, and for
+n = 4 denote by J(C) = 1600(c0 − A0) the normalized function obtained
+from 2.9 with j = 0. Then it can be seen that although each term in the
+expression of J(C) has weight four, we have
+J(Q) = f ′(z)4J(C) − 200h′(z)
+h(z) f ′(z)3I4(C),
+clearly showing that the function J is not a semi-invariant.
+8
+
+4. Infinitesimal generators of the induced group action
+The equivalence group G in (18) of the general linear equation (4) induces
+another Lie pseudo group Gc acting on the coefficients of (4) [3]. For linear
+equations with maximal symmetries, their most general form depends as
+already noted on only two arbitrary functions, instead of n. For instance,
+the most general form of linear equations of order four admitting a maximal
+symmetry algebra is given on account of (15) by
+y(4) + c3y(3) + c2y′′ + 1
+8
+�
+4c2c3 − c3
+3 + 8c′
+2 − 6c3c′
+3 − 4c′′
+3
+�
+y′
++
+1
+1600
+�
+144c2
+2 − 11c4
+3 + 400c3c′
+2 − 288c2
+3c′
+3 − 336c′2
+3
+− 8c2
+�
+c2
+3 + 4c′
+3
+�
++ 480c′′
+2 − 560c3c′′
+3 − 320c(3)
+3
+�
+y = 0
+(23)
+and it is expressible solely in terms of the coefficients cn−1 and cn−2, here
+c3 and c2.
+Although Eq.
+(23) is a very special case of the general Eq.
+(4), its
+equivalence group is the same group G in (18) because equivalent equations
+have similar symmetry groups. Consequently the infinitesimal generators
+X0 of the group Gc for (4) will also be valid for equations with maximal
+symmetries. In particular to obtain the specific infinitesimal generators for
+equations with maximal symmetries expressed only in terms of the two arbi-
+trary functions, it will be sufficient to substitute the characterizing equations
+(10) into the expression for X0.
+A method for finding the infinitesimal generator X0 has been proposed
+in [8]. If we denote by
+X = ξ ∂x + η ∂y + φn−1 ∂cn−1 + · · · + φ0 ∂c0
+(24)
+the infinitesimal generator of (4) in which the coefficients
+C = (c0, c1, . . . , cn−1)
+are also considered as dependent variables, then the method of [8] consists of
+finding a set of minimum conditions for which the projection V = ξ ∂x +η ∂y
+of X on the (x, y)-space reduces to the infinitesimal generator V 0 =
+�
+ξ0, η0�
+of the equivalence group G.
+This set of minimal conditions imposed to
+φ = (φ0, φ1, . . . , φn−1) yields a function φ0 = (φ0
+0, φ0
+1, . . . , φ0
+n−1) so that the
+expression for X0 takes the form
+X0 = ξ0 ∂x + φ0
+n−1 ∂cn−1 + · · · + φ0
+0 ∂c0.
+(25)
+9
+
+In practice, the determination of the symmetry generator X for the general
+linear equation (4) is computationally exhaustive, and a popular Lie sym-
+metry software such as MathLie (See [11]) computes X only for n ≤ 4 due to
+computer memory problems (on an Intel Core2 Quad CPU machine) while
+another well-known similar Lie symmetry software such as SYM [12] does
+not compute symmetries such as X that involve several dependent variables
+for a single independent variable.
+We therefore need an efficient simplifying ansatz for the manual compu-
+tation of X0 at orders higher than the fourth. For this, we note that as the
+full symmetry group of (4) with C considered also as dependent variable
+should leave the equation invariant, the transformation of the dependent
+and the independent variables should preserve the form of the equation, ex-
+cept for the introduction of a constant term independent of y which should
+be offset by the subsequent transformations of the coefficient C. This means
+that in (24), we must have
+ξ = f(x),
+η = g(x)y + h(x).
+(26)
+A verification of (26) is possible by direct calculation for equations of order
+not higher than the fourth using the MathLie software, while for orders
+higher than four, the validity of the generators X and X0 found can be
+tested through the satisfaction of the infinitesimal condition of invariance
+applied to the general linear equation (4), and to the semi-invariants In found
+in (22), respectively. Recall that the infinitesimal criterion of invariance for
+the infinitesimal generator X of (4) is given by
+X[n] �
+∆(x, y(n); C)
+�
+= 0,
+whenever ∆(x, y(n); C) = 0,
+(27)
+where X[n] represents the n-th prolongation of X. Regarding the verification
+of the infinitesimal condition of invariance for semi-invariants, we note that
+if for some group element α ∈ Gc we set Q = α·C, then every semi-invariant
+of Gc satisfies F(α · C) = w(α) · F(C) for some weight function w, and X0
+is an infinitesimal generator of Gc if and only if
+X0 · F = −dw(e)F,
+for all such functions F, where w(e) is the differential of w at the identity
+element e of Gc. In the actual case of (4) and Gc (which is the same as G
+except that it acts on the space of coefficients), for α ≡ (f, g) specified in
+(18) we have w(α) = f ′(z)3, and for each generator X0 ≡ X0(n) found, it
+is readily verified that
+X0 · In = −3f ′(x)In,
+(28)
+10
+
+as required.
+To our knowledge the infinitesimal generators X0 of the induced pseudo
+group Gc has been computed only for third order equations, or for the nor-
+mal or the Laguerre-Forsyth forms of equations of low orders not exceeding
+five [13, 14, 9]. This is due in part as already mentioned to the intensive
+computational requirements for the calculation of these generators, but also
+because the more systematic method for finding them proposed in [8] is
+relatively recent.
+We list in the next three theorems the general expressions for the in-
+finitesimal generators Xn of GS and X0
+n of Gc and for the three most general
+canonical forms of linear equations, where the subscript n denotes the order
+of the equation.
+Theorem 1. For the general linear equation of order n in standard form
+(4), the infinitesimal generators Xn of GS and X0
+n of Gc have the following
+expressions, where f, g and h are arbitrary functions of x, and δk
+0 denotes
+the Kronecker delta.
+a)
+Xn = f∂x + (yg + h) ∂y +
+n−1
+�
+k=0
+Φn
+k∂ck,
+(29a)
+where
+Φn
+k = −(n − k)ckf ′ +
+n−k
+�
+j=1
+ck+j
+��k + j
+j + 1
+�
+f (j+1) −
+�k + j
+j
+�
+g(j)
+�
++ δk
+0
+
+−ck
+h
+y +
+n−k
+�
+j=1
+ck+j
+�k + j
+j
+�h(j)
+y
+
+ ,
+for k = 0, . . . , n − 1.
+(29b)
+b)
+X0
+n = f∂x +
+n−1
+�
+k=0
+Φn
+k∂ck,
+(30a)
+11
+
+where
+Φn
+k = −(n − k)ckf ′ +
+n−k
+�
+j=1
+�
+−
+�k + j
+j
+�
+g(j) +
+�k + j
+j + 1
+�
+f (j+1)
+�
+ck+j,
+for k = 0, . . . , n − 1.
+(30b)
+Proof. We let the generator Xn be in the form
+Xn =ξ∂x + η∂y +
+n−1
+�
+k=0
+Φn
+k∂ck,
+(31)
+where the functions ξ, η, and Φn
+k are to be specified. We know from the
+ansatz (26) that ξ = f(x) and η = g(x)y + h(x) for some arbitrary func-
+tions f, g and h of x. The prolongation formula for X[n]
+n
+is well-known [15].
+Writing down this expression and applying the infinitesimal condition of in-
+variance (27) gives the usual determining equations for the coefficients ξ, η
+and Φn
+k. Although the procedure is a lengthy one, thanks to the ansatz (26)
+these determining equations are easily solved and lead to the expressions in
+(29).
+For the second part of the theorem, the result follows by noting that
+according to the algorithm of [8] already described for finding X0
+n, one es-
+sentially only need to find the minimum set of conditions which reduce the
+projection {f(x), g(x)y + h(x)} of Xn onto the (x, y)-space to the infinitesi-
+mal generator of the equivalence group. From the expressions of the equiv-
+alence transformations given in (18), it follows that the required minimal
+set of condition reduces to {h = 0}. Applying these conditions to (29) and
+dropping the term in ∂y gives the required expression (30).
+Theorem 2. For the general linear equation in reduced normal form, i.e.
+in the form (4) with cn−1 = 0, the generators Xn of GS and X0
+n of Gc have
+the following expressions, in terms of the arbitrary functions f and h of x.
+a)
+Xn = f∂x +
+�
+y
+��n − 1
+2
+�
+f ′ + K1
+�
++ h
+�
+∂y +
+n−2
+�
+k−0
+Φn
+k∂ck,
+(32a)
+12
+
+where
+Φn
+k = −(n − k)f ′ck +
+n−k
+�
+j=1
+ck+j
+��k + j
+j + 1
+�
+−
+�k + j
+j
+�n − 1
+2
+�
+f (j+1)
++ δk
+0
+
+−ck
+h
+y +
+n−k
+�
+j=1
+ck+j
+�k + j
+j
+�h(j)
+y
+
+ ,
+for k = 0, . . . , n − 2.
+(32b)
+b)
+X0
+n = f∂x +
+n−2
+�
+k=0
+Φn
+k∂ck,
+(33a)
+where
+Φn
+k = −(n − k)ckf ′ +
+n−k
+�
+j=1
+ak+j
+��k + j
+j + 1
+�
+−
+�k + j
+j
+�n − 1
+2
+�
+f (j+1),
+for k = 0, . . . , n − 2.
+(33b)
+Proof. The expressions for Xn and X0
+n are to be sought in the form (29)
+and (30), respectively, as the normal form of (4) is a special case of that
+equation. The main difference is that the equivalence transformations for
+the normal form are no longer given by (18) but by the much restricted
+version
+x =T(z),
+y = λ
+�
+T ′(z)
+� n−1
+2 w(z)
+(34)
+where T is an arbitrary function and λ an arbitrary constant.
+This has
+infinitesimal generator
+V =f(x)∂x + y
+�n − 1
+2
+f ′(x) + k1
+�
+∂y,
+(35)
+where f is an arbitrary function and k1 an arbitrary constant. Since the func-
+tions f and g in (29) and (30) are precisely the parameters of the infinitesimal
+generator of the equivalence group, to obtain (32) and (33), we only need to
+replace g in the latter expressions by the substitution g = n−1
+2 f ′ + k1 and
+to drop the term in cn−1. This yields (32) and (33).
+13
+
+The Laguerre-Forsyth form of the general linear equation is the equation
+of the form (4) in which the coefficients cn−1 and cn−2 of terms of second
+and third highest orders have vanished. In principle, such a transformation
+can be realized by means of the change of variables of the form
+{z, x} =
+12
+n(n − 1)(n + 1)cn−2,
+y = w exp
+�
+− 1
+n
+� z
+z0
+cn−1dx
+�
+,
+(36a)
+where
+{z, x} =
+�
+z′z(3) − (3/2)z′′2�
+z′ −2
+(36b)
+is the Schwarzian derivative, and z′ = dz/dx. The Laguerre-Forsyth form of
+(4) is therefore of an implicit nature in the sense that (36) can not always
+be solved explicitly for z. Nevertheless, such a form is still of interest, in
+particular because linear equations often occur in this form.
+Theorem 3. For the general linear equation (4) in Laguerre-Forsyth form,
+the infinitesimal generators Xn of GS and X0
+n of Gc have the following ex-
+pressions, where a0, a1, a2, and k1 are arbitrary constants, and h an arbitrary
+function.
+a)
+Xn = (a2x2 + a1x + a0)∂x +
+�
+y
+�
+k1 + n − 1
+2
+�
+2a2x + a1
+��
++ h
+�
+∂y
++
+n−3
+�
+k=0
+�
+− (n − k)(2a2x + a1)ck + a2(k + 1)(k + 1 − n)ck+1
++ δk
+0
+�
+− ck
+h
+y +
+n−k
+�
+j=1
+�k + j
+j
+�h(j)
+y
+��
+∂ck
+(37)
+b)
+X0
+n =
+�
+a2x2 + a1x + a0
+�
+∂x
++
+n−3
+�
+k=0
+[−(n − k)(2a2x + a1)ck + a2(k + 1)(k + 1 − n)ck+1] ∂ck.
+(38)
+14
+
+Proof. As in the proof of Theorem 2, we only need to note that as the
+Laguerre-Forsyth form is a special case of the normal form, its generators
+Xn and X0
+n should be sought in the form (32) and (33), respectively. More
+exactly, we only need to find the specific expression for the parameter f of the
+equivalence transformation corresponding to the Laguerre-Forsyth form and
+substitute this into (32) and (33), and to drop the term involving cn−2 in the
+resulting expressions. It is well-known that the equivalence transformations
+of the Laguerre-Forsyth form of (4) are invertible transformations of the form
+(34) in which T(z) is a linear fractional transformation. The corresponding
+infinitesimal generator is thus of the form (35), in which f(x) = a2x2+a1x+
+a0, for some arbitrary constants a2, a1, and a0. This is the expression for f
+which was to be found, and this completes the proof.
+Thanks to the ansatz (26) a direct computation of Xn and X0
+n for equa-
+tions of low orders up to seven has been performed and confirms the validity
+of the expressions given in the three preceding theorems. It should also be
+noted that unlike the case of equations in standard or in normal forms, the
+generator X0
+n of Gc in the case of the Laguerre-Forsyth form involves only a
+finite number of constant parameters. This means that the invariant func-
+tions for this form of the general linear equation are much easier to compute,
+as already noted by Forsyth [10] who obtained an expression for them by a
+direct analysis.
+As noted earlier, for equations with a maximal symmetry algebra which
+are already expressed solely in terms of the two coefficients cn−1 and cn−2, to
+obtain the corresponding infinitesimal generator X0, it suffices to substitute
+in the expression for X0
+n corresponding to the general linear equation (4)
+the corresponding characterizing equations which give an expression for the
+other coefficients solely in terms of cn−1 and cn−2 alone. For instance, for
+n = 4, the expression for X0
+n corresponding to the normalized equation (23)
+15
+
+has, on account of (15) and (29), an expression given by
+ξ = f
+φ0
+3 = −c3f ′ − 4g′ + 6f ′′
+φ0
+2 = −2c2f ′ − 3c3g′ + 3c3f ′′ − 6g′′ + 4f (3)
+φ0
+1 = 3
+8f ′(c3
+3 − 8c′
+2 + 6c3c′
+3 + 4c′′
+3) − 3c3g′′ + c3f (3)
++ c2
+�
+−3
+2c3f ′ − 2g′ + f ′′
+�
+− 4g(3) + f (4)
+φ0
+0 = −1
+8g′(8c′
+2 − c3(−4c2 + c2
+3 + 6c′
+3) − 4c′′
+3) − c2g′′ − c3g(3)
+− g(4) −
+1
+400f ′�
+144c2
+2 − 11c4
+3 − 288c2
+3c′
+3 − 8c2(c2
+3 + 4c′
+3)
+− 80c3(5c′
+2 − 7c′′
+3) + 16(21c′2
+3 − 30c′′
+2 + 20c(3)
+3 )
+�
+.
+(39)
+5. Concluding remarks
+We reiterate the fact already mentioned that the symmetry properties
+obtained in this paper for linear equations also apply to the infinite dimen-
+sional class of nonlinear equations which are equivalent to a given linear
+equation admitting a maximal symmetry algebra. For instance, in the sim-
+plest case of the free fall equation y′′ = 0, an invertible point transformation
+of the form x = f(z, w), y = g(z, w) shows that the most general class of
+second order (linear or nonlinear) equations admitting a maximal symmetry
+algebra has the form
+fzgz,z − gzfz,z + w3
+z (−gwfw,w + fwgw,w)
++ w2
+z (−gzfw,w − 2gwfz,w + fzgw,w + 2fwgz,w)
++ wz (−2gzfz,w − gwfz,z + 2fzgz,w + fwgz,z) + (fzgw − fwgz) wz,z = 0.
+Moreover, linearization methods under point transformations are available
+for odes of order up to three [5, 6], and this is very meaningful as for practical
+considerations most odes of physical relevance fall within this range.
+One of the most interesting properties of linear equations with maximal
+symmetries is that their solution can be obtained by a very simple super-
+position formula from that of the second order source equation [2]. More
+specifically, thanks to (7), any such equation can always be assumed to be in
+the normal reduced form (5). In particular, the corresponding second order
+source equation has the form y′′ + by = 0, for a certain function b = b(x). If
+16
+
+we let u and v be two linearly independent solutions of this source equation,
+then n linearly independent solutions of an equation of the form (5) with
+the same source equation are given by
+yk = ukvn−1−k,
+k = 0, . . . , n − 1.
+The latter fact can be used not only for finding analytic solutions of nonlinear
+equations, but also in the test of numerical schemes. Indeed, when testing
+a numerical scheme, it is always helpful to have an appropriate collection
+of nonlinear problems for which one or more explicit analytic solutions are
+available [16, 17].
+The infinitesimal generators X0
+n of the induced pseudo group of transfor-
+mations Gc found in Section 4 are of a more general interest. One of their
+main role is in the determination of the invariants (and semi-invariants) of
+the family of equations, and these functions can in turn be used for a com-
+plete classification of the given family of equations [18, 19], thus reducing
+the study in each equivalence class to that of the canonical equation. For a
+much practical and immediate use, they are very efficient in testing whether
+a given function is an invariant of the related family of equation, and any
+given invariant of the family can also easily be used to test some necessary
+conditions of equivalence between two given equations.
+References
+[1] S. Lie, Klassification und Integration von gew¨ohnlichen Differentialgle-
+ichungen zwischen x, y, die eine Gruppe von Transformationen gestet-
+ten. I, Math. Ann. 22 (1888) 213–253.
+[2] J. Krause, L. Michel, Equations diff´erentielles lin´eaires d’ordre n > 2
+ayant une alg`ebre de Lie de sym´etrie de dimension n + 4, C.R. Acad.
+Sci. Paris 307 (1988) 905–910.
+[3] S.
+Lie,
+Theorie der Transformationsgruppen,
+Dritter
+Abschnitt,
+Abteilun. I. Unter Mitwirkung von Pr. F. Engel, Teubner, Leipzig, 1893.
+[4] E. Laguerre, Sur les ´equations diff´erentielles lin´eaires du troisi`eme ordre,
+C.R. Acad. Sci. Paris 88 (1879) 116–119.
+[5] F.M. Mahomed, P.G.L. Leach, Symmetry Lie Algebras of nth Order
+Ordinary Differential Equations, J. Math. Anal. Appl. 151 (1990) 80–
+107.
+17
+
+[6] N.H. Ibragimov, F. Magri, Geometric proof of Lie’s linearization theo-
+rem, Nonlinear Dynam. 36 (2004) 41–46.
+[7] J.C. Ndogmo, F.M. Mahomed, On certain properties of linear it-
+erative equations, Cent. Eur. J. Math. 12 no. 4, (2014) 648–657,
+arXiv:1207.6851.
+[8] J.C. Ndogmo, A method for the equivalence group and its infinitesimal
+generators, J. Phys. A: Math. Theor. 41 (2008) 102001.
+[9] J.C. Ndogmo, Generating Relative and Absolute Invariants of Linear
+Differential Equations, Int. Math. Forum 4 (2009) 873–886.
+[10] A.R. Forsyth, Invariants, covariants, and quotient-derivatives associ-
+ated with linear differential equations, Philos. Trans. R. Soc. Lond. 179
+(1888) 377–489.
+[11] G. Baumann, Symmetry Analysis of Differential Equations with Math-
+ematica, Springer, New York, 2000.
+[12] S. Dimas D. Tsoubelis, SYM: A new symmetry–finding package for
+Mathematica, in: N.H. Ibragimov, C. Sophocleous, P.A. Damianou
+(Eds.), Proceedings of 10th International Conference in Modern Group
+Analysis, Larnaca, Cyprus, 2004, pp 64–70.
+[13] N.H. Ibragimov, Infinitesimal method in the theory of invariants of
+algebraic and differential equations, Not. S. Afr. Math. Soc. 29 (1997)
+61–70.
+[14] J.C. Ndogmo, On structure-preserving point transformations of differ-
+ential equations, Phys. Lett. A 373 (2009) 1226–1232.
+[15] P.J. Olver, Applications of Lie Groups to Differential Equations,
+Springer, New York, 1986.
+[16] B. Bradie, A Friendly Introduction to Numerical Analysis, Prentice-
+Hall, Upper Saddle River, 2006.
+[17] N.J. Higham, Accuracy and Stability of Numerical Algorithms, Second
+Edition, SIAM, Philadelphia, 2002.
+[18] M. Fels, P.J. Olver, Moving coframes. II. Regularization and theoretical
+foundations, Acta. Appl. Math. 55 (1999) 127–208.
+[19] O.I. Morozov, Contact-equivalence problem for linear hyperbolic equa-
+tions, J. Math Sci. (N.Y.) 135 (2006) 2680–2694.
+18
+

9dAyT4oBgHgl3EQfqPin/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf,len=403
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='00540v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='CA]  2 Jan 2023  Coefficient characterization of linear differential equations  with maximal symmetries  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Ndogmo  School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050,  South Africa  Abstract  A characterization of the general linear equation in standard form admit-  ting a maximal symmetry algebra is obtained in terms of a simple set of  conditions relating the coefficients of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' As a consequence, it is  shown that in its general form such an equation can be expressed in terms of  only two arbitrary functions, and its connection with the Laguerre-Forsyth  form is clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The characterizing conditions are also used to derive an  infinite family of semi-invariants, each corresponding to an arbitrary order  of the linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Finally a simplifying ansatz is established, which  allows an easier determination of the infinitesimal generators of the induced  pseudo group of equivalence transformations, for all the three most general  canonical forms of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Keywords:  Coefficient characterization, maximal symmetry algebra,  canonical form, induced equivalence group, infinitesimal generators  2010 MSC: 70G65, 34C20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Introduction  By a result of Lie [1], a linear ordinary differential equation (ode) of a  general order n is known to have a symmetry algebra of maximal dimension  dn if it is reducible by a point transformation to the equation y(n) = 0, which  will henceforth be referred to as the canonical form of the linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  In a much recent paper Krause and Michel [2] proved the converse of this  result and also showed that a linear equation is iterative if and only if its  symmetry algebra has the maximal dimension dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' (By the cited result of Lie  Email address: jean-claude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='ndogmo@wits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='za (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Ndogmo)  [1], dn = n+4 for n ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Characterizing linear equations having a symmetry  algebra of maximal dimension is therefore the same as characterizing linear  equations that are reducible by a point transformation to the canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  The latter characterization for the third-order equation y(3) + c2 y′′ + c1 y′ +  c0 y = 0 is due to Lie [3] and Laguerre [4] who showed independently that  this equation is reducible to the canonical form if and only if its coefficients  satisfy the equation  54c0 − 18c1c2 + 4c3  2 − 27c′  1 + 18c2c′  2 + 9c′′  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (1)  This characterization also clearly applies to all nonlinear odes which are  linearizable by point transformations [5, 6], as the latter transformations do  not alter the dimension of the symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  In this paper, we extend this characterization to equations of higher or-  ders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' It turns out that for each equation of order n there will be n − 2  characterizing equations, and the limitation of our presentation of the char-  acterizing equations only up to the order five is simpy due to their very  large size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' However, we give a description of the method for deriving this  characterization for equations of any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The derivation of these char-  acterizing equations is also based on the canonical normal form of linear  equations admitting a maximal symmetry algebra that was obtained in [5]  from a symmetry approach, and in [7] from an iterative approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' These  characterizing equations therefore also represent a generalization of the re-  sults of [5] and [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' We then deduce that the most general form of a linear  equation admitting a maximal symmetry algebra can be expressed in stan-  dard form in terms of only two arbitrary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' We also deduce that the  Laguerre-Forsyth form of a linear equation reduces to the canonical form if  and only if the equation has maximal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Although we do not give the characterizing equations for each linear  equation of order n, we note however that among the n − 2 characterizing  equations exactly one of them represents a semi-invariant of the equation,  that is a function of the coefficients of the equation whose expression does  not change when the dependent variable is transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  We obtain an  expression for these semi-invariants for equations of all orders and describe  some of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Finally, using some simplifying assumptions and the method of [8], we  give expressions for both the symmetry generator Xn of GS and X0  n of the  induced pseudo group of transformations Gc, and for all three most general  canonical forms of linear equations of a general order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Here, GS denotes  the symmetry group of the general linear equation in which the arbitrary  2  functions are considered as additional dependent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Coefficient characterization  A method based on a symmetry approach has been proposed in [5] for  characterizing the coefficients of linear ordinary differential equations (odes)  that admit a maximal symmetry algebra, but only for equations in reduced  normal form (in which the term of second highest order vanishes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' In a more  recent paper [7] a similar characterization based on an iterative approach was  proposed, in which according to a result of Krause and Michel [2] a linear  equation admitting a maximal symmetry is simply viewed as an iterative  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' By iterative equation, we mean an equation of the form  Ψn[y] = 0,  y = y(x),  n ≥ 1  (2a)  where  Ψ1[y] = ry′ + sy,  Ψn[y] = Ψn−1 [Ψ[y]] ,  (2b)  and where r = r(x) and s = s(x) are the parameters of the source equation  Ψ1[y] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' This characterization shows that in its reduced normal form, a  general linear equation depends solely on one arbitrary function a = a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  For equations of orders three to five, the corresponding equations are given  as follows:  y(3) + ay′ + a′  2 y = 0  (3a)  y(4) + ay′′ + a′y′ +  � 3  10a′′ +  9  100a2  �  y = 0  (3b)  y(5) + ay(3) + 3  2a′y′′ +  � 9  10a′′ + 16  100a2  �  y′ +  �1  5a(3) + 16  100aa′  �  y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (3c)  However, as a linear equation need not occur in its reduced normal form,  but rather in the most general standard form, it is thus useful to obtain the  corresponding characterization for equations in standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' We let the  general linear equation be given in standard form as  ∆(x, y(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' C) ≡ y(n) + cn−1 y(n−1) + cn−2 y(n−2) + · · · + c0 y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (4)  3  where C = (c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , cn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Suppose that such an equation has a symmetry  algebra of maximal dimension and let its corresponding reduced normal form  be given by  y(n) + Bn−2 y(n−2) + Bn−3 y(n−3) + · · · + B0 y = 0,  (5)  where the Bj for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n−2 are its coefficients and depend as already  noted above on a single arbitrary function a = Bn−2 and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Let  y(n) + An−1 y(n−1) + An−2 y(n−2) + · · · + A0 y = 0  (6)  be the corresponding standard form of (5), which may be obtained by a  transformation of the form  y �→ ye− 1  n  � x  x0 An−1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (7)  Then (4) and (6) must be identical, and in particular the nonzero coef-  ficient An−1 introduced by the transformation (7) satisfies An−1 = cn−1,  and more generally we have  cj = Aj,  for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (8)  Note that the coefficients cj in (4) are mere symbols and we wish to find a  relationship among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Given that in (5) the function Bn−2 is precisely  the arbitrary function a(x) labeling the equation, it can be shown by a  recursive procedure, or even by induction on n that  An−2 = a + n − 1  2n  c2  n−1 + n − 2  2  c′  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Therefore, solving the equation cn−2 = An−2 for a gives  a = cn−2 −  �n − 1  2n  c2  n−1 + n − 2  2  c′  n−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (9)  Consequently, the characterizing equations for linear equations in standard  form with maximal symmetry algebra are given by the remaining n − 2  equations  cj = Aj,  j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n − 3,  (10)  in which the function a and its derivatives are substituted with the corre-  sponding expressions given by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' If a linear equation in standard form (4) has maximal sym-  metry, then in its general form it may be expressed in terms of only two ar-  bitrary functions, namely the functions cn−1 and cn−2, and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The result readily follows from the fact that the functions Aj in (10)  then depend only on a and its derivatives, while (9) shows that the function  a depends precisely on cn−1, cn−2, and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' A linear equation in standard form (4) with cn−1 = cn−2 = 0  has maximal symmetry algebra if and only if cj = 0 for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  In other  words a linear equation has maximal symmetry algebra if and only if its  Laguerre-Forsyth form corresponds to the canonical equation y(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' After all a Laguerre transformation is also a point transformation  although it cannot always be explicitly constructed for a given equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Since equations equivalent under point transformation have similar Lie al-  gebras, it readily follows that if the Laguerre-Forsyth form of an equation is  y(n) = 0, then the equation has maximal symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The converse  of the corollary is a direct application of proposition 1, and the fact that in  (10) the cj turn out to be polynomial functions with no constant terms of  cn−1, cn−2, and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  As an immediate consequence of the corollary, linear equations such as  y(3) +f(x)y = 0 or y(4)+f(x)y′ = 0 have maximal symmetry algebras if and  only if the function f(x) vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' We now make use of (10) and  (9) to explicitly derive the characterizing equations for maximal symmetry  algebras for equations of orders three to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  For n = 3, it is readily found that in (6) we have  A0 = 1  54  �  18ac2 + 2c3  2 + 27a′ + 18c2c′  2 + 18c′′  2  �  ,  (11)  while the corresponding expression for a in (9) reduces to  a = c1 −  �c2  2  3 + c′  2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (12)  Applying (12) into (11) and substituting the resulting expression for A0  into (10) gives exactly the already cited equation (1) found by Lie [3] and  Laguerre [4] and given by  54c0 − 18c1c2 + 4c3  2 − 27c′  1 + 18c2c′  2 + 9c′′  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  The most general form of a linear third-order equation admitting a maximal  symmetry algebra can thus be expressed in terms of only two arbitrary  functions c1(x) and c2(x) in the form of  y(3) + c2 y′′ + c1 y′ + 1  54  �  18c1c2 − 4c3  2 + 27c′  1 − 18c2c′  2 − 9c′′  2  �  y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (13)  5  Equation (13) naturally reduces to (3a) for c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  For n = 4, we successively get  a = 1  8  �  8c2 − 3c2  3 − 12c′  3  �  (14a)  A1 = 1  2  �  ac3 + c3  3  16 + a′ + 3  4c3c′  3 + c′′  3  �  (14b)  6400A0 = 576a2 + 400a(c2  3 + 4c′  3)  + 5  �  5c4  3 + 120c2  3c′  3 + 320c3(a′ + c′′  3)  �  + 80  �  15c′2  3 + 24a′′ + 20c(3)  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (14c)  Substituting (14a) into (14b) and (14c) gives the two equations  8c1+ = 4c2c3 − c3  3 + 8c′  2 − 6c3c′  3 − 4c′′  3  (15a)  1600c0 = 144c2  2 − 11c4  3 + 400c3c′  2 − 288c2  3c′  3 − 336c′2  3  − 8c2(c2  3 + 4c′  3) + 480c′′  2 − 560c3c′′  3 − 320c(3)  3  (15b)  which represent the characterizing equations for maximal symmetry algebra  for equations of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Note that conversely any linear fourth order equa-  tion whose coefficients satisfy (15) must be iterative, which is why conditions  such as (15) are termed characterizing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' if the coefficients  of a fourth order equation of the form (4) satisfy (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' then its reduced nor-  mal form has,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' after the substitution of the expressions for c0 and c1 given  by (5) in terms of c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' and their derivatives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' the form  w(4) + Q2w′′ + Q1w′ + Q0w = 0  (16a)  where  Q2 = c2 − 3  8(c2  3 + 4c′  3)  (16b)  Q1 = c′  2 − 3  4(c3c′  3 + 2c′′  3)  (16c)  Q0 =  3  6400(192c2  2 + 27c4  3 − 48c′2  3 − 144c2(c2  3 + 4c′  3))  +  3  6400(27c4  3 + 216c2  3c′  3 + 640c′′  2 − 480c3c′′  3 − 960c′′′  3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (16d)  The coefficients Qj thus obtained clearly satisfy the conditions  Q1 = Q′  2  and  Q0 = ( 3  10Q′′  2 +  9  100Q2  2)  6  prescribed by (3b) for iterative equations, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  For equations of order n = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' by proceeding as above for the orders three  and four,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' we obtain the following n − 2 = 3 characterizing equations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='c2 = (30c3c4 − 8c3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 + 75c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3 − 60c4c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 + 50c′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4)/50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='(17a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='1250 c1 = +200c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3 − 18c4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 + 750c4c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3 − 580c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 − 850c′2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='− 10c3(c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 + 5c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4) + 1125c′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3 − 1400c4c′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 − 1000c(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='(17b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='6250 c0 = 200c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3c4 + 14c5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 − 25c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3 + 40c3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='− 125c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 − 750c4c′2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 + 1125c4c′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3 − 850c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4c′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='− 2750c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4c′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 + 1250c(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='− 2000c4c(3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='− 1250c(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='− 10c3(11c3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 + 100c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='3 − 85c4c′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4 − 75c′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (17c)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Semi-invariants of linear equations  The group of equivalence transformations of the general linear equation  (4) is given by invertible point transformations of the form  x = f(z),  y = g(z)w(z),  (18)  and they preserve the linearity and the homogeneity of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Let  w(n) + Qn−1 w(n−1) + Qn−2 w(n−2) + · · · + Q0 w = 0  (19)  be the transformed version of (4) under (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' By a semi-invariant of (4)  we shall mean a function F = F(c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , cn−1) of the coefficients of the  equation which have the same expression for the transformed equation when  the dependent variable (alone) changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' It is well known that under (18)  the expression of the semi-invariant for the transformed equation is related  to that for the original equation [9, 10] by the equality  F(Q0, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , Qn−1) =  �dx  dz  �µ  F(c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , cn−1),  (20)  where µ is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  In this case we say that the semi-variant F has  index µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' To each expression of the form dkcj/dxk, let us assign the weight  (n − j) + k, and we let this weight function be multiplicative so that the  product cpcq has weight (n − p) + (n − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' It is well known that for a given  semi-invariant all terms have the same weight and that this weight coincides  with the index of the semi-invariant [9, 10] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  7  A closer look at the set of characterizing equations (10) shows that pre-  cisely one of them corresponds to a semi-invariant of the equation, namely  the relation cn−3 = An−3, which gives rise to the semi-invariant F =  An−3 − cn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  First of all, using the method of either [7] or [5], it can be proved that  the coefficient Bn−3 in (5) satisfies Bn−3 = n−2  2 a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Consequently, using the  expression of the function a in (9) it follows by induction on n that the  coefficient An−3 in (6) is given by  An−3 =n − 2  n  cn−1cn−2 − (n − 1)(n − 2)  3n2  c3  n−1 + n − 2  2  c′  n−2  − (n − 1)(n − 2)  2n  cn−1c′  n−1 − (n − 1)(n − 2)  12  c′′  n−1,  (21)  so that the corresponding invariant function In has expression  In =n − 2  n  cn−1cn−2 − (n − 1)(n − 2)  3n2  c3  n−1 + n − 2  2  c′  n−2  − (n − 1)(n − 2)  2n  cn−1c′  n−1 − (n − 1)(n − 2)  12  c′′  n−1 − cn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (22)  The fact that the function In = In(c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , cn−1) in (22) is a semi-invariant  can readily be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' First each term in this expression has weight three,  and we readily see that  In(Q0, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , Qn−1) = f ′(z)3In(c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , cn−1),  which proves the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Although the invariant functions In in (22) are originally defined only for  n ≥ 3, their expression shows that they vanish identically for n = 1 or n = 2,  by letting cj = 0 for j < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' This vanishing can be interpreted by the fact that  all first order and all second order linear equations are all equivalent through  a point transformation to the equations y′ = 0 and y′′ = 0, respectively, and  therefore they do not have nontrivial invariant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  On the other hand it should be noted that the other equations in the  characterizing system (10) do not give rise to invariant functions except for  the value j = n − 3 in that system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Indeed, denote collectively  by C and Q the coefficients in equations (4) and (19), respectively, and for  n = 4 denote by J(C) = 1600(c0 − A0) the normalized function obtained  from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='9 with j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Then it can be seen that although each term in the  expression of J(C) has weight four, we have  J(Q) = f ′(z)4J(C) − 200h′(z)  h(z) f ′(z)3I4(C),  clearly showing that the function J is not a semi-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Infinitesimal generators of the induced group action  The equivalence group G in (18) of the general linear equation (4) induces  another Lie pseudo group Gc acting on the coefficients of (4) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For linear  equations with maximal symmetries, their most general form depends as  already noted on only two arbitrary functions, instead of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For instance,  the most general form of linear equations of order four admitting a maximal  symmetry algebra is given on account of (15) by  y(4) + c3y(3) + c2y′′ + 1  8  �  4c2c3 − c3  3 + 8c′  2 − 6c3c′  3 − 4c′′  3  �  y′  +  1  1600  �  144c2  2 − 11c4  3 + 400c3c′  2 − 288c2  3c′  3 − 336c′2  3  − 8c2  �  c2  3 + 4c′  3  �  + 480c′′  2 − 560c3c′′  3 − 320c(3)  3  �  y = 0  (23)  and it is expressible solely in terms of the coefficients cn−1 and cn−2, here  c3 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Although Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (23) is a very special case of the general Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (4), its  equivalence group is the same group G in (18) because equivalent equations  have similar symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Consequently the infinitesimal generators  X0 of the group Gc for (4) will also be valid for equations with maximal  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' In particular to obtain the specific infinitesimal generators for  equations with maximal symmetries expressed only in terms of the two arbi-  trary functions, it will be sufficient to substitute the characterizing equations  (10) into the expression for X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  A method for finding the infinitesimal generator X0 has been proposed  in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' If we denote by  X = ξ ∂x + η ∂y + φn−1 ∂cn−1 + · · · + φ0 ∂c0  (24)  the infinitesimal generator of (4) in which the coefficients  C = (c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , cn−1)  are also considered as dependent variables, then the method of [8] consists of  finding a set of minimum conditions for which the projection V = ξ ∂x +η ∂y  of X on the (x, y)-space reduces to the infinitesimal generator V 0 =  �  ξ0, η0�  of the equivalence group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  This set of minimal conditions imposed to  φ = (φ0, φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , φn−1) yields a function φ0 = (φ0  0, φ0  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , φ0  n−1) so that the  expression for X0 takes the form  X0 = ξ0 ∂x + φ0  n−1 ∂cn−1 + · · · + φ0  0 ∂c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (25)  9  In practice, the determination of the symmetry generator X for the general  linear equation (4) is computationally exhaustive, and a popular Lie sym-  metry software such as MathLie (See [11]) computes X only for n ≤ 4 due to  computer memory problems (on an Intel Core2 Quad CPU machine) while  another well-known similar Lie symmetry software such as SYM [12] does  not compute symmetries such as X that involve several dependent variables  for a single independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  We therefore need an efficient simplifying ansatz for the manual compu-  tation of X0 at orders higher than the fourth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For this, we note that as the  full symmetry group of (4) with C considered also as dependent variable  should leave the equation invariant, the transformation of the dependent  and the independent variables should preserve the form of the equation, ex-  cept for the introduction of a constant term independent of y which should  be offset by the subsequent transformations of the coefficient C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' This means  that in (24), we must have  ξ = f(x),  η = g(x)y + h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (26)  A verification of (26) is possible by direct calculation for equations of order  not higher than the fourth using the MathLie software, while for orders  higher than four, the validity of the generators X and X0 found can be  tested through the satisfaction of the infinitesimal condition of invariance  applied to the general linear equation (4), and to the semi-invariants In found  in (22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Recall that the infinitesimal criterion of invariance for  the infinitesimal generator X of (4) is given by  X[n] �  ∆(x, y(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' C)  �  = 0,  whenever ∆(x, y(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' C) = 0,  (27)  where X[n] represents the n-th prolongation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Regarding the verification  of the infinitesimal condition of invariance for semi-invariants, we note that  if for some group element α ∈ Gc we set Q = α·C, then every semi-invariant  of Gc satisfies F(α · C) = w(α) · F(C) for some weight function w, and X0  is an infinitesimal generator of Gc if and only if  X0 · F = −dw(e)F,  for all such functions F, where w(e) is the differential of w at the identity  element e of Gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' In the actual case of (4) and Gc (which is the same as G  except that it acts on the space of coefficients), for α ≡ (f, g) specified in  (18) we have w(α) = f ′(z)3, and for each generator X0 ≡ X0(n) found, it  is readily verified that  X0 · In = −3f ′(x)In,  (28)  10  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  To our knowledge the infinitesimal generators X0 of the induced pseudo  group Gc has been computed only for third order equations, or for the nor-  mal or the Laguerre-Forsyth forms of equations of low orders not exceeding  five [13, 14, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' This is due in part as already mentioned to the intensive  computational requirements for the calculation of these generators, but also  because the more systematic method for finding them proposed in [8] is  relatively recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  We list in the next three theorems the general expressions for the in-  finitesimal generators Xn of GS and X0  n of Gc and for the three most general  canonical forms of linear equations, where the subscript n denotes the order  of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For the general linear equation of order n in standard form  (4), the infinitesimal generators Xn of GS and X0  n of Gc have the following  expressions, where f, g and h are arbitrary functions of x, and δk  0 denotes  the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  a)  Xn = f∂x + (yg + h) ∂y +  n−1  �  k=0  Φn  k∂ck,  (29a)  where  Φn  k = −(n − k)ckf ′ +  n−k  �  j=1  ck+j  ��k + j  j + 1  �  f (j+1) −  �k + j  j  �  g(j)  �  + δk  0  \uf8ee  \uf8f0−ck  h  y +  n−k  �  j=1  ck+j  �k + j  j  �h(j)  y  \uf8f9  \uf8fb ,  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (29b)  b)  X0  n = f∂x +  n−1  �  k=0  Φn  k∂ck,  (30a)  11  where  Φn  k = −(n − k)ckf ′ +  n−k  �  j=1  �  −  �k + j  j  �  g(j) +  �k + j  j + 1  �  f (j+1)  �  ck+j,  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (30b)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' We let the generator Xn be in the form  Xn =ξ∂x + η∂y +  n−1  �  k=0  Φn  k∂ck,  (31)  where the functions ξ, η, and Φn  k are to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' We know from the  ansatz (26) that ξ = f(x) and η = g(x)y + h(x) for some arbitrary func-  tions f, g and h of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The prolongation formula for X[n]  n  is well-known [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Writing down this expression and applying the infinitesimal condition of in-  variance (27) gives the usual determining equations for the coefficients ξ, η  and Φn  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Although the procedure is a lengthy one, thanks to the ansatz (26)  these determining equations are easily solved and lead to the expressions in  (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  For the second part of the theorem, the result follows by noting that  according to the algorithm of [8] already described for finding X0  n, one es-  sentially only need to find the minimum set of conditions which reduce the  projection {f(x), g(x)y + h(x)} of Xn onto the (x, y)-space to the infinitesi-  mal generator of the equivalence group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' From the expressions of the equiv-  alence transformations given in (18), it follows that the required minimal  set of condition reduces to {h = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Applying these conditions to (29) and  dropping the term in ∂y gives the required expression (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For the general linear equation in reduced normal form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  in the form (4) with cn−1 = 0, the generators Xn of GS and X0  n of Gc have  the following expressions, in terms of the arbitrary functions f and h of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  a)  Xn = f∂x +  �  y  ��n − 1  2  �  f ′ + K1  �  + h  �  ∂y +  n−2  �  k−0  Φn  k∂ck,  (32a)  12  where  Φn  k = −(n − k)f ′ck +  n−k  �  j=1  ck+j  ��k + j  j + 1  �  −  �k + j  j  �n − 1  2  �  f (j+1)  + δk  0  \uf8ee  \uf8f0−ck  h  y +  n−k  �  j=1  ck+j  �k + j  j  �h(j)  y  \uf8f9  \uf8fb ,  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (32b)  b)  X0  n = f∂x +  n−2  �  k=0  Φn  k∂ck,  (33a)  where  Φn  k = −(n − k)ckf ′ +  n−k  �  j=1  ak+j  ��k + j  j + 1  �  −  �k + j  j  �n − 1  2  �  f (j+1),  for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (33b)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The expressions for Xn and X0  n are to be sought in the form (29)  and (30), respectively, as the normal form of (4) is a special case of that  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The main difference is that the equivalence transformations for  the normal form are no longer given by (18) but by the much restricted  version  x =T(z),  y = λ  �  T ′(z)  � n−1  2 w(z)  (34)  where T is an arbitrary function and λ an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  This has  infinitesimal generator  V =f(x)∂x + y  �n − 1  2  f ′(x) + k1  �  ∂y,  (35)  where f is an arbitrary function and k1 an arbitrary constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Since the func-  tions f and g in (29) and (30) are precisely the parameters of the infinitesimal  generator of the equivalence group, to obtain (32) and (33), we only need to  replace g in the latter expressions by the substitution g = n−1  2 f ′ + k1 and  to drop the term in cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' This yields (32) and (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  13  The Laguerre-Forsyth form of the general linear equation is the equation  of the form (4) in which the coefficients cn−1 and cn−2 of terms of second  and third highest orders have vanished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' In principle, such a transformation  can be realized by means of the change of variables of the form  {z, x} =  12  n(n − 1)(n + 1)cn−2,  y = w exp  �  − 1  n  � z  z0  cn−1dx  �  ,  (36a)  where  {z, x} =  �  z′z(3) − (3/2)z′′2�  z′ −2  (36b)  is the Schwarzian derivative, and z′ = dz/dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The Laguerre-Forsyth form of  (4) is therefore of an implicit nature in the sense that (36) can not always  be solved explicitly for z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Nevertheless, such a form is still of interest, in  particular because linear equations often occur in this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For the general linear equation (4) in Laguerre-Forsyth form,  the infinitesimal generators Xn of GS and X0  n of Gc have the following ex-  pressions, where a0, a1, a2, and k1 are arbitrary constants, and h an arbitrary  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  a)  Xn = (a2x2 + a1x + a0)∂x +  �  y  �  k1 + n − 1  2  �  2a2x + a1  ��  + h  �  ∂y  +  n−3  �  k=0  �  − (n − k)(2a2x + a1)ck + a2(k + 1)(k + 1 − n)ck+1  + δk  0  �  − ck  h  y +  n−k  �  j=1  �k + j  j  �h(j)  y  ��  ∂ck  (37)  b)  X0  n =  �  a2x2 + a1x + a0  �  ∂x  +  n−3  �  k=0  [−(n − k)(2a2x + a1)ck + a2(k + 1)(k + 1 − n)ck+1] ∂ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (38)  14  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' As in the proof of Theorem 2, we only need to note that as the  Laguerre-Forsyth form is a special case of the normal form, its generators  Xn and X0  n should be sought in the form (32) and (33), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' More  exactly, we only need to find the specific expression for the parameter f of the  equivalence transformation corresponding to the Laguerre-Forsyth form and  substitute this into (32) and (33), and to drop the term involving cn−2 in the  resulting expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' It is well-known that the equivalence transformations  of the Laguerre-Forsyth form of (4) are invertible transformations of the form  (34) in which T(z) is a linear fractional transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' The corresponding  infinitesimal generator is thus of the form (35), in which f(x) = a2x2+a1x+  a0, for some arbitrary constants a2, a1, and a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' This is the expression for f  which was to be found, and this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Thanks to the ansatz (26) a direct computation of Xn and X0  n for equa-  tions of low orders up to seven has been performed and confirms the validity  of the expressions given in the three preceding theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' It should also be  noted that unlike the case of equations in standard or in normal forms, the  generator X0  n of Gc in the case of the Laguerre-Forsyth form involves only a  finite number of constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' This means that the invariant func-  tions for this form of the general linear equation are much easier to compute,  as already noted by Forsyth [10] who obtained an expression for them by a  direct analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  As noted earlier, for equations with a maximal symmetry algebra which  are already expressed solely in terms of the two coefficients cn−1 and cn−2, to  obtain the corresponding infinitesimal generator X0, it suffices to substitute  in the expression for X0  n corresponding to the general linear equation (4)  the corresponding characterizing equations which give an expression for the  other coefficients solely in terms of cn−1 and cn−2 alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' for  n = 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' the expression for X0  n corresponding to the normalized equation (23)  15  has,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' on account of (15) and (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' an expression given by  ξ = f  φ0  3 = −c3f ′ − 4g′ + 6f ′′  φ0  2 = −2c2f ′ − 3c3g′ + 3c3f ′′ − 6g′′ + 4f (3)  φ0  1 = 3  8f ′(c3  3 − 8c′  2 + 6c3c′  3 + 4c′′  3) − 3c3g′′ + c3f (3)  + c2  �  −3  2c3f ′ − 2g′ + f ′′  �  − 4g(3) + f (4)  φ0  0 = −1  8g′(8c′  2 − c3(−4c2 + c2  3 + 6c′  3) − 4c′′  3) − c2g′′ − c3g(3)  − g(4) −  1  400f ′�  144c2  2 − 11c4  3 − 288c2  3c′  3 − 8c2(c2  3 + 4c′  3)  − 80c3(5c′  2 − 7c′′  3) + 16(21c′2  3 − 30c′′  2 + 20c(3)  3 )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  (39)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Concluding remarks  We reiterate the fact already mentioned that the symmetry properties  obtained in this paper for linear equations also apply to the infinite dimen-  sional class of nonlinear equations which are equivalent to a given linear  equation admitting a maximal symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For instance, in the sim-  plest case of the free fall equation y′′ = 0, an invertible point transformation  of the form x = f(z, w), y = g(z, w) shows that the most general class of  second order (linear or nonlinear) equations admitting a maximal symmetry  algebra has the form  fzgz,z − gzfz,z + w3  z (−gwfw,w + fwgw,w)  + w2  z (−gzfw,w − 2gwfz,w + fzgw,w + 2fwgz,w)  + wz (−2gzfz,w − gwfz,z + 2fzgz,w + fwgz,z) + (fzgw − fwgz) wz,z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  Moreover, linearization methods under point transformations are available  for odes of order up to three [5, 6], and this is very meaningful as for practical  considerations most odes of physical relevance fall within this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  One of the most interesting properties of linear equations with maximal  symmetries is that their solution can be obtained by a very simple super-  position formula from that of the second order source equation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' More  specifically, thanks to (7), any such equation can always be assumed to be in  the normal reduced form (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' In particular, the corresponding second order  source equation has the form y′′ + by = 0, for a certain function b = b(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' If  16  we let u and v be two linearly independent solutions of this source equation,  then n linearly independent solutions of an equation of the form (5) with  the same source equation are given by  yk = ukvn−1−k,  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  The latter fact can be used not only for finding analytic solutions of nonlinear  equations, but also in the test of numerical schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Indeed, when testing  a numerical scheme, it is always helpful to have an appropriate collection  of nonlinear problems for which one or more explicit analytic solutions are  available [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='  The infinitesimal generators X0  n of the induced pseudo group of transfor-  mations Gc found in Section 4 are of a more general interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' One of their  main role is in the determination of the invariants (and semi-invariants) of  the family of equations, and these functions can in turn be used for a com-  plete classification of the given family of equations [18, 19], thus reducing  the study in each equivalence class to that of the canonical equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' For a  much practical and immediate use, they are very efficient in testing whether  a given function is an invariant of the related family of equation, and any  given invariant of the family can also easily be used to test some necessary  conditions of equivalence between two given equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Mahomed, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
+page_content=' Leach, Symmetry Lie Algebras of nth Order  Ordinary Differential Equations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfqPin/content/2301.00540v1.pdf'}
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@@ -0,0 +1,1887 @@
+SlideVQA: A Dataset for Document Visual Question Answering on Multiple
+Images
+Ryota Tanaka, Kyosuke Nishida, Kosuke Nishida, Taku Hasegawa, Itsumi Saito, Kuniko Saito
+NTT Human Informatics Laboratories, NTT Corporation
+{ryouta.tanaka.rg, kyosuke.nishida.rx, kosuke.nishida.ap, taku.hasegawa.ps, itsumi.saito.df, kuniko.saito.ku}@hco.ntt.co.jp
+Abstract
+Visual question answering on document images that con-
+tain textual, visual, and layout information, called document
+VQA, has received much attention recently. Although many
+datasets have been proposed for developing document VQA
+systems, most of the existing datasets focus on understand-
+ing the content relationships within a single image and not
+across multiple images. In this study, we propose a new multi-
+image document VQA dataset, SlideVQA, containing 2.6k+
+slide decks composed of 52k+ slide images and 14.5k ques-
+tions about a slide deck. SlideVQA requires complex rea-
+soning, including single-hop, multi-hop, and numerical rea-
+soning, and also provides annotated arithmetic expressions
+of numerical answers for enhancing the ability of numerical
+reasoning. Moreover, we developed a new end-to-end docu-
+ment VQA model that treats evidence selection and question
+answering in a unified sequence-to-sequence format. Exper-
+iments on SlideVQA show that our model outperformed ex-
+isting state-of-the-art QA models, but that it still has a large
+gap behind human performance. We believe that our dataset
+will facilitate research on document VQA.
+Introduction
+Building intelligent agents that can read and comprehend
+real-world documents, such as webpages, office documents,
+lecture slides, etc., has been a long-standing goal of artificial
+intelligence. To achieve this goal, machine reading compre-
+hension (MRC), a central task in natural language under-
+standing, has been intensively studied. The typical defini-
+tion of the MRC task is quite simple, wherein given a short
+natural language text as a context and a question about it,
+a machine reads the text and then answers the question by
+extracting a span from the text (Rajpurkar et al. 2016; Ra-
+jpurkar, Jia, and Liang 2018). However, this definition is far
+from real-world applications, such as customer service chat-
+bots on e-commerce websites (Cui et al. 2017) and assis-
+tant systems for reading professional literature (Hong et al.
+2019), in that the context is composed entirely of text, with
+no graphical elements.
+To this end, visual question answering on document im-
+ages (document VQA) has received much attention. It is a
+challenging vision and language task that requires methods
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+to reason about document layout, textual content, and visual
+elements (Mathew, Karatzas, and Jawahar 2021; Tanaka,
+Nishida, and Yoshida 2021; Mathew et al. 2022). When
+the primary content in a document is text (e.g., e-mails
+and forms) and the task is to understand it on the basis of
+its layout information, state-of-the-art models have already
+achieved nearly human-level performance (Xu et al. 2021;
+Powalski et al. 2021). On the other hand, challenges remain
+when it comes to handling diverse real-world documents.
+First and foremost is that current models are not capable of
+performing reasoning across multiple images since the ex-
+isting datasets focus on testing reasoning ability on a single
+image. Moreover, compared with humans, document VQA
+models still have trouble understanding documents that con-
+tain visual elements and understanding questions that re-
+quire numerical reasoning (Mathew et al. 2022).
+To address the above challenges, we introduce a new doc-
+ument VQA dataset1, SlideVQA, for tasks wherein given a
+slide deck composed of multiple slide images and a corre-
+sponding question, a system selects a set of evidence im-
+ages and answers the question. Slide decks are one of the
+most efficient document types that arrange visual and textual
+elements for communication. As shown in Figure 1, Slide-
+VQA requires complex reasoning over slide images, includ-
+ing single-hop, multi-hop, and numerical reasoning. These
+reasoning skills play essential roles in MRC tasks (Yang
+et al. 2018; Dua et al. 2019).
+Our main contributions are summarized as follows:
+• We introduce a novel task and dataset, SlideVQA,
+wherein to answer its questions, a machine has to read
+and comprehend a slide deck. It is the largest multi-
+image document VQA dataset containing 2.6k+ slide
+decks (each consisting of 20 slides) and 14.5k questions.
+It also provides bounding boxes around textual and visual
+elements for understanding document layout and arith-
+metic expressions for numerical reasoning.
+• We developed a Multi-Modal Multi-image Document
+VQA model, M3D, to jointly perform evidence selection
+and question answering tasks and to enhance numerical
+reasoning by generating arithmetic expressions.
+1Our dataset and codes are publicly available at https://github.
+com/nttmdlab-nlp/SlideVQA
+arXiv:2301.04883v1  [cs.CL]  12 Jan 2023
+
+…
+p.4
+p.11
+p.12
+Q: What is the difference in the competition media percent
+age between East and the region with 12% of journalists?
+A: 5% (11% – 6% )
+Evidence pages:  4, 12
+Answer type: Non-Span  Reasoning type: Multi-hop, Numerical
+Q: What is the percentage of the internal meeting decision?
+Q: What is the tip-off media percentage in the region with 
+70% of journalists and South?
+A: 13%, 16%                       Evidence pages:  4, 12
+Answer type: Multi-Span   Reasoning type: Multi-hop
+A:  21%      
+Evidence pages:  11
+Answer type: Single-Span   Reasoning type: Sing-hop
+10
+THE FIRST STEP TO THE BIG STORY
+The research sheds light on how journalists conceive story ideas. Internal 
+meetings, tip-offs, events and primary research were the most popular 
+sources with 63 percent of journalists relying on these activities for story 
+ideas. Internal brainstorm sessions and editorial meetings were found to be 
+the most preferred sources for generating fresh content-related ideas. Online 
+content and social networks seem to be triggers for the same pie of journalists 
+across all experience levels.
+In an informal interview chat, one of the journalists said that reading and 
+surfing could provide some cues, but that it was sheer hard work when one 
+finally wrote a story. There was no way “one could do desktop stories’’, said 
+another journalist. Yet, another journalist felt that the Net could provide a 
+trigger. Seasoned journalists, more often than not, develop sustainable 
+relationships with their sources, consult experts and interview key people to 
+get the flavour for the subjects they are reporting on.
+Looking specifically at regional variations in story conceptualization, more 
+journalists from the South look for story triggers in competitive media vis-à-
+vis other regions. The regional analysis also indicated that most journalists 
+from East draw on events to evolve fresh story ideas. The popularity of 
+interactive formats provides an immense opportunity for corporates to reach 
+out to media in the East through press events. 
+SECTION 1
+Internal brainstorming 
+meetings are the 
+biggest source of story 
+ideas #mediainsights
+In terms of getting story 
+ideas, age is no bar as 
+far as reliance on online 
+media is concerned 
+#mediainsights
+News hooks across 
+competitive media 
+serve as story idea 
+triggers for 16% of 
+journalists in the South, 
+versus 9% in the North 
+#mediainsights
+Events are more 
+favored by journalists 
+in the East, followed 
+by the North, West and 
+South #mediainsights
+Communications agencies 
+are most preferred by 
+journalists covering sports, 
+followed by those covering 
+Business & Corporate and 
+Science & Technology 
+#mediainsights
+Women reporters have a 
+greater affinity for 
+communications agencies 
+versus their male 
+counterparts 
+#mediainsights
+11
+Competition media/
+channel/newspaper
+10%
+Tip-off
+14%
+An event
+15%
+Social Network
+07%
+Online content/news
+08%
+09%
+Primary research
+13%
+Others
+03%
+Communication 
+agencies
+THE FIRST STEP TO THE BIG STORY
+Internal meeting 
+decision
+21%
+SECTION 1
+1 2
+THE FIRST STEP TO THE BIG STORY
+Internal meeting decision
+Competition media
+Tip-off
+Communication agencies
+Primary research
+Others
+An event
+Social Network
+Online content
+North
+South
+East
+West
+20%
+9%
+13%
+16%
+8%
+8%
+9%
+13%
+4%
+26%
+16%
+16%
+7%
+2%
+10%
+10%
+10%
+3%
+29%
+6%
+15%
+20%
+3%
+3%
+6%
+18%
+0%
+20%
+11%
+14%
+14%
+5%
+6%
+8%
+19%
+3%
+SECTION 1
+1 3
+THE FIRST STEP TO THE BIG STORY
+Business & 
+Corporate
+Lifestyle & 
+Entertainment
+Science & 
+Tech
+Sports
+21%
+10%
+13%
+12%
+5%
+13%
+10%
+13%
+3%
+25%
+7%
+14%
+16%
+8%
+6%
+10%
+13%
+1%
+19%
+11%
+10%
+17%
+11%
+11%
+8%
+9%
+4%
+19%
+9%
+13%
+19%
+8%
+3%
+13%
+14%
+2%
+Internal meeting decision
+Competition media
+Tip-off
+Communication agencies
+Primary research
+Others
+An event
+Social Network
+Online content
+SECTION 1
+Figure 1: Examples from our SlideVQA dataset. Some questions can be answered through single-hop, multi-hop, and numerical
+reasoning. The colors of the words match the image borders with the same colors. (·) of the right example in the answer denotes
+an annotated arithmetic expression to derive the final answer. The slide deck can be viewed at https://www.slideshare.net/
+mslgroup/mediainsights-evolving-sources-of-news-for-media.
+• Our model outperformed existing state-of-the-art QA
+models on SlideVQA, but its performance is still below
+that of humans by a large margin.
+Related Work
+Datasets for VQA on document images.
+Document
+VQA is the task of answering questions about document
+images, and some useful datasets have been published,
+such as DocVQA (Mathew, Karatzas, and Jawahar 2021),
+VisualMRC (Tanaka, Nishida, and Yoshida 2021), Web-
+SRC (Chen et al. 2021), and InfographicVQA (Mathew et al.
+2022). The task assumes that the datasets have a single rele-
+vant image, containing all the facts required to answer.
+The work most related to ours is DocCVQA (Tito,
+Karatzas, and Valveny 2021), wherein a large collection of
+document images is used to answer a given question. Our
+dataset differs from DocCVQA, as follows. First, Slide-
+VQA consists of 14.5k questions, wheres DocCVQA pro-
+vides only 20 questions. Second, SlideVQA requires multi-
+hop reasoning over multiple slides to find the answer, while
+DocCVQA requires only single-hop reasoning on individual
+images to find the answer. Besides these differences, Slide-
+VQA provides questions that require numerical reasoning
+and arithmetic expression annotations to answer numerical
+questions (e.g., “30 - 28” for the answer “2”): no other VQA
+dataset, including InfographicVQA that requires numerical
+reasoning, provides such annotations. Furthermore, Slide-
+VQA provides the largest number of bounding boxes on all
+of the collected images among the related datasets.
+Document VQA Models.
+In parallel with the develop-
+ment of datasets, Transformer (Vaswani et al. 2017) has
+come to be used for understanding unstructured text in docu-
+ment images. LayoutLM (Xu et al. 2020), LayoutLMv2 (Xu
+et al. 2021), LayoutT5 (Tanaka, Nishida, and Yoshida 2021),
+and TILT (Powalski et al. 2021) have achieved impressive
+results in single-image document VQA tasks by combining
+textual, layout, and visual features. By contrast, we focus on
+endowing models with the ability to reason and comprehend
+multiple images. Moreover, while Tito, Karatzas, and Val-
+veny (2021) used a pipeline of retrieval and reading models
+for DocCVQA, we use multi-task learning that jointly per-
+forms evidence selection and question answering.
+Multi-modal question answering.
+This type takes textual
+and visual information as input contexts, which is different
+from document VQA that takes only a document image as
+the input context. TQA (Kembhavi et al. 2017) is comprised
+of middle-school science lessons containing diagrams and
+text. MultiModalQA (Talmor et al. 2021) requires joint rea-
+soning over text, tables, and images in Wikipedia.
+VQA on videos or image sets.
+VideoQA focuses on an-
+swering questions about video frames of TV shows (Lei
+et al. 2018, 2020) and movies (Tapaswi et al. 2016). A simi-
+lar task is VQA on image sets (ISVQA), which involves han-
+dling photos taken from different viewpoint indoors (Bansal,
+Zhang, and Chellappa 2020). By contrast, our dataset also
+requires a model to understand the text in images.
+Slide
+images
+understanding.
+Monica
+Haurilet
+and
+Stiefelhagen (2019); Haurilet et al. (2019) introduced a
+benchmark for object segmentation on slide-pages. Sun
+et al. (2021); Fu et al. (2022) tackled the task of generating
+slides from research papers. Our work is the first to focus
+on answering questions on sets of slide images.
+Reasoning over textual documents.
+Numerical reason-
+ing plays an important role in NLP tasks (Dua et al. 2019;
+Zhang et al. 2020, 2021). Moreover, multi-hop reasoning has
+taken the spotlight as it aligns with the multi-hop nature of
+how humans reason to acquire knowledge, and has led to a
+
+20:20MSLROREWORD
+CETMOE
+XOOTTHEFRSTSTEPTOTHEBOSTORY
+GEXPERTSPEAKTotal=309
+vrs.
+10
+5-20
+BATI
+responden
+%ZE
+215
+North
+70
+South
+REGIONDataset
+Document
+Multi-images Multi-hop Numerical
+Answer
+Document images #QAs #Images #BBoxes #Arithmetic #Evidence
+source
+input
+reasoning
+reasoning
+type
+modal type
+annotations candidates
+DocVQA
+industry
+SS
+TL
+50k
+12k
+–
+–
+1
+VisualMRC
+web-pages
+Ab
+TLV
+30k
+10k
+64k
+–
+1
+WebSRC
+web-pages
+SS
+TLV
+400k
+6.4k
+–
+–
+1
+InfographicVQA infographics
+✓
+SS, MS, NS
+TLV
+30k
+5k
+–
+–
+1
+DocCVQA
+industry
+✓
+MS
+TL
+0.02k
+14k
+–
+–
+14k
+SlideVQA (Ours)
+slide decks
+✓
+✓
+✓
+SS, MS, NS
+TLV
+14.5k
+52k
+890k
+1.7k
+20
+Table 1: Comparison of question answering datasets on document images. Answer types can be broken down into abstractive
+(Ab), single-span (SS), multi-span (MS), and non-span (NS). “T/L/V” denotes the “text/layout/visual” modality of images.
+proliferation of benchmarks (Talmor and Berant 2018; Yang
+et al. 2018). However, there is as yet no dataset for devel-
+oping models to perform both multi-hop and numerical rea-
+soning on document images.
+The SlideVQA Task and Dataset
+Task Overview and Formulation
+The SlideVQA task, requires a system to answer a question
+about a slide deck, which is composed of an ordered set of
+slide images and to select evidence slide images. We formu-
+late the end-to-end SlideVQA task as follows:
+MAINTASK (SlideVQA). Given a question q and a slide
+deck I = {I1, . . . , IK} (K = 20), a model outputs an an-
+swer y and selects relevant slides ˆI = {ˆI1, . . . , ˆIK′}.
+The task can be decomposed into two subtasks:
+SUBTASK 1 (Evidence Selection). Given a question q and a
+slide deck I, a model identifies the images ˆI from which to
+derive the answer y.
+SUBTASK 2 (Question Answering). Given a question q and
+the slide images (I or ˆI), a model outputs an answer y.
+SlideVQA has three answer types (see the examples in
+Figure 1). A single-span answer is a contiguous sequence of
+tokens in the reading order extracted from the image, and a
+multi-span answer is formed from multiple spans from the
+image. A non-span answer is not extracted and is composed
+of numerical values and visual appearances.
+We can also use annotations of bounding boxes around
+the objects (and their categories) to understand the seman-
+tic structure of images and annotations of arithmetic expres-
+sions to understand numerical reasoning as additional input
+at training. These annotations are not given at inference.
+Dataset Collection
+In this section, we describe the collection process of the
+SlideVQA dataset. To control the annotation quality, we re-
+cruited crowd workers located in English-speaking countries
+and who had passed a rigorous qualification procedure. Ad-
+ditionally, we asked other workers to assess the quality of
+the annotated samples after each collection step.
+Slide decks collection.
+First, we selected and downloaded
+25,327 slide decks composed of more than 20 slides from
+slideshare2 and covering 39 topics. We kept the first 20 slides
+2https://www.slideshare.net/
+Figure 2: Example of collected bounding boxes. Colored
+boxes and words were annotated by workers. The image can
+be viewed at https://www.slideshare.net/andrybrewok/big-
+data-analytics-a-social-network-approach.
+and truncated the rest of the pages. Then, the workers filtered
+the collected decks that did not meet the following criteria:
+(i) the main language is English; (ii) the content is easy for
+workers to understand; (iii) the decks must contain one or
+more graphs, tables, figures, or numerical data to avoid cre-
+ating questions requiring only text-level understanding.
+Bounding boxes and categories annotation.
+To facilitate
+understanding of the semantic components of images, we
+annotated all images with bounding boxes and their cate-
+gories. The workers indicated specific objects in each image
+by annotating bounding boxes around the objects and classi-
+fying them into nine classes that were based on SPaSe (Mon-
+ica Haurilet and Stiefelhagen 2019) as follows:
+• Title: presentation title, slide title
+• Page-text: text in slide, bullet-point text list, text list
+• Obj-text: text in a figure, image, diagram or table
+• Caption: description of figure, image, diagram, or table
+• Other-text: footnote, date, affiliation, code, URL
+• Diagram: a graphical representation of data, a process
+• Table: data arranged in rows and columns
+• Image: drawing, logo, map, screenshot, realistic image
+• Figure: graph with data points and coordinates
+As shown in Figure 2, SlideVQA provides densely anno-
+tated bounding boxes in images.
+
+Title
+RESEARCHROADMAP
+Diagram
+Obj-text
+Online Data
+Obi-text
+SocialNetwork
+Obi-text
+Obi-text
+SCBDResearch
+Obi-text
+StructtiredData
+DataMliningandPatternRecognition
+Obi-text
+Obi-text
+ConversationalData
+Sentirrent.Analysis
+Caption
+GOAL descriptions,predictions,optimisation and simulation
+Captian
+arta.marketing,communications,knowiedge
+management,operations,finance,etcTitle
+Page-text
+Obj-text
+Caption
+Other-text
+Diagram
+Table
+Image
+Figure
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Percentage of images (%)
+Text
+Layout
+Visual
+(a) Bounding box categories.
+Single-Hop
+Multi-Hop
+Single-Hop
+& Numerical
+Multi-Hop
+& Numerical
+0
+10
+20
+30
+40
+50
+Percentage of questions (%)
+(b) Reasoning types.
+Arithmetic
+Count
+Comparison
+0
+10
+20
+30
+40
+50
+Percentage of numerical reasoning questions (%)
+(c) Numerical operation types.
+Single-Span
+Multi-Span
+Non-Span
+0
+10
+20
+30
+40
+50
+60
+70
+Percentage of answers (%)
+(d) Answer types.
+Figure 3: Distribution of bounding box categories, reasoning
+types, numerical operations, and answer types in the test set.
+Single-hop QA creation.
+We asked the workers to create
+12,466 QA pairs by selecting a single slide image from a
+slide deck. The selected slide can be used as evidence to
+tell whether a system arrived at the right answer for the
+right reasons. We encouraged questions that needed numeri-
+cal reasoning, including operations of arithmetic expressions
+with {+, −, /, ∗}, counting, and comparisons. Additionally,
+the workers avoided creating questions that (i) contained se-
+lected page numbers; (ii) required external knowledge; (iii)
+were common to all of the slides (e.g., “What is the title?”).
+Multi-hop questions creation.
+We created 2,018 QA
+pairs for multi-hop reasoning by editing the single-hop ques-
+tions created in the previous step. For example at the left
+of Figure 1, “North” is replaced by the phrase “the re-
+gion with 70% of journals”. To this end, we first identified
+one or two bridge entities in the created questions, and the
+workers selected related slides as evidence that mentioned
+the identified ones. Then, the content of the selected slides
+was utilized to replace the entities in the created questions.
+The process of creating multi-hop questions by editing may
+produce unnatural questions, as mentioned in the “Limita-
+tions” section, but is easily scalable. A similar approach was
+taken with MultiModalQA (Talmor et al. 2021), which re-
+quires multi-hop reasoning over text, tables, and images in
+Wikipedia.
+Arithmetic expression annotation.
+We provided arith-
+metic expressions like “30 - 28” in which the final numerical
+answer can be arrived at with the four arithmetic operations.
+The interpretation of the answer generation process is im-
+portant for creating explainable QA models.
+what
+which
+how
+in
+regarding
+on
+who
+is
+when
+according
+where
+are
+was
+the
+were
+did
+do
+by
+looking
+approximately
+at
+over
+for
+between
+does
+if
+as
+during
+have
+is
+are
+percentage
+was
+does
+type
+comes
+two
+kind
+three
+percent
+do
+year
+follows
+did
+four
+were
+city
+country
+position
+happens
+category
+level
+makes
+step
+the
+has
+should
+age
+android
+animal
+apparatus
+binds
+car
+color
+contains
+directly
+languages
+new
+part
+particle
+share
+smartphone
+types
+country
+is
+has
+was
+type
+are
+company
+region
+of
+year
+age
+passenger
+team
+two
+requires
+brand
+category
+day
+group
+position
+state
+achieved
+animated
+app
+area
+bank
+coffee
+frp
+geographic
+market
+part
+performs
+political
+reason
+renewable
+republic
+route
+segment
+seven
+store
+three
+vehicle
+website
+many
+much
+does
+is
+large
+what
+the
+which
+how
+was
+gaap
+oceania
+the
+mhealth
+google
+ccd
+customers
+denmark
+top
+buy
+europe
+the
+which
+what
+is
+wrote
+are
+invented
+investigated
+the
+there
+that
+a
+an
+breaking
+differentiation
+was
+did
+is
+this
+to
+is
+does
+there
+the
+more
+most
+there
+the
+a
+employee
+percentage
+presentation
+there
+more
+truck
+brazil
+more
+profit
+the
+more
+people
+updates
+what
+at
+what
+how
+how
+which
+the
+time
+which
+and
+an
+you
+of
+wine
+how
+belongs
+is
+created
+core
+than
+is
+the
+a
+another
+an
+growth
+on
+the
+two
+three
+four
+examples
+five
+six
+some
+of
+is
+was
+responded
+very
+the
+basf
+the
+a
+of
+between
+after
+items
+of
+types
+of
+is
+the
+the
+is
+does
+comes
+difference
+has
+had
+the
+higher
+greater
+more
+a
+more
+of
+the
+accounts
+the
+group
+vehicle
+more
+people
+types
+billions
+steps
+total
+employees
+points
+reasons
+stages
+years
+did
+greater
+has
+is
+more
+does
+was
+the
+year
+was
+percentage
+year
+country
+year
+many
+global
+installed
+example
+key
+smartphone
+u.s.
+app
+market
+customers
+of
+what
+search
+slide
+day
+the
+the
+the
+more
+which
+the
+the
+the
+the
+the
+more
+more
+percentage
+the
+Figure 4: Distribution of the first three words of the ques-
+tions.
+Statistics and Analysis
+SlideVQA contains 14,484 QA pairs from 2,619 slide decks,
+consisting of 52,480 slide images annotated with 890,945
+bounding boxes. We split the dataset into 10,617 questions
+for training, 1,652 (2,215) questions for development (test),
+making sure that each deck appears in the same split.
+Images.
+SlideVQA provides the largest number of images
+covering broad range of topics among the datasets shown
+in Table 1. Moreover, SlideVQA provides the largest num-
+ber of bounding box annotations, where the number of the
+annotations in SlideVQA is 14.7 times that of VisualMRC.
+Figure 3a shows the distribution of bounding boxes broken
+down into nine categories, which cover all classes, including
+visually related ones (Image and Figure), unlike DocVQA
+and DocCVQA. To analyze the OCR tokens, we extracted
+the text shown in the images by using the Google Cloud Vi-
+sion API3. As a result, the number of OCR tokens the sys-
+tem should consider simultaneously is larger (1488.88 to-
+kens) than those of single-image document VQA datasets;
+the largest dataset (InfographicVQA) has 217.89 tokens.
+Questions and answers.
+As shown in Table 1, SlideVQA
+requires complex reasoning including single/multi-hop, and
+numerical reasoning. Figure 3b shows the diverse distribu-
+tion of questions related to reasoning types. 49.3% of the
+questions require multi-hop or numerical reasoning. More-
+over, SlideVQA provides annotations of arithmetic expres-
+sions to improve numerical reasoning. Figure 3c shows the
+distribution of numerical operations. 25.5% of the numerical
+questions require arithmetic operations, which current sys-
+tems have particular difficulty answering. Figure 3d shows
+that multi-span and non-span account for 32.4% of the an-
+swers, indicating systems also need to generate answers as
+well as extract multiple spans.
+Figure 4 shows the sunburst pattern of the first three words
+of the questions. “In” and “Regarding” are frequent first
+3https://cloud.google.com/vision
+
+Task prefix (𝒕): 
+{“Evidence Selection”,
+“Question Answering”}
+Question (𝒒)
+Slide-1
+Slide-2
+Slide-𝐾
+Slide deck (𝑰):
+Input features
+extraction 
+Input sequence 𝑥!
+Input sequence	𝑥"
+Input sequence	𝑥#
+Multi-modal
+Encoder
+Evidence 
+Selector
+Answer/Arithmetic-
+expression Decoder
+Answer: Steve Jobs
+Expression: 30 - 28
+or
+Evidence pages: 2, 4
+Calculator
+2
+Token
+Segment
+Layout
+Visual
+Task prefix + Question 
++ Page number
+Slide image
++
++
++
+Object detection & OCR
+Q
+Q
+Q
+Title
+Title
+Image
+[Title]
+THE
+…
+…
+…
+…
+[Image]
+task
+:
+…
+(b) Input sequence and embeddings
+(a) Our M3D modules
+Figure 5: (a) Our encoder-decoder model architecture and (b) input representations. Given a question with a task prefix and
+a slide deck, the model outputs a corresponding answer/arithmetic-expression and evidence pages. The calculator outputs the
+final answer to calculate the generated arithmetic expression.
+words because SlideVQA needs to search for evidence im-
+ages from a slide deck, which is a special pattern in multi-
+text document QA (Yang et al. 2018).
+Our Model
+Figure 5 shows an overview of our model, called M3D
+(Multi-Modal Multi-image Document VQA model). We use
+Fusion-in-Decoder (FiD) (Izacard and Grave 2021), which is
+a state-of-the-art multi-text encoder-decoder model, as our
+base model and initialize FiD with a pre-trained T5 (Raf-
+fel et al. 2020). We extend FiD to perform the end-to-end
+SlideVQA task (defined in MAINTASK) by (i) performing
+evidence selection and question answering tasks as a unified
+sequence-to-sequence format using multi-task learning, (ii)
+predicting arithmetic expressions as intermediate reasoning
+steps instead of generating answers directly to enhance nu-
+merical reasoning, and (iii) modifying the input sequence to
+learn the visual layout and content of the image.
+Multi-modal Task-Specific Input
+Input token sequence.
+For each image Ik, we first use
+Faster-RCNN (Ren et al. 2015), which was trained on Slide-
+VQA, to extract N semantic regions (bounding boxes) and
+their labels (e.g., Title and Image). We parse the slide im-
+age for each extracted region r by using an OCR engine and
+apply a sub-word tokenizer to obtain OCR tokens Wr
+k =
+{wr
+k,1, . . . , wr
+k,n} and corresponding OCR bounding boxes.
+To jointly train the evidence selection and question answer-
+ing tasks, we add different task prefixes t ∈ {Evidence
+Selection, Question Answering} to the encoder
+input. Specifically, the input sequence is as follows:
+xk = (task:t question:q page:ek context:ck),
+where the sequence concatenates each slide and page num-
+ber pair (ck, ek) with the question q and task prefix t. To tell
+the role of each region, we insert region labels [Rri
+k ], cor-
+responding to the region label of the i-th region ri in k-th
+page, before the OCR tokens Wri
+k extracted in ri:
+ck = ([Rr1
+k ], Wr1
+k , [Rr2
+k ], Wr2
+k , . . . , [RrN
+k ], WrN
+k )
+Input embedding.
+Following LayoutT5 (Tanaka, Nishida,
+and Yoshida 2021), the input embeddings z of the encoder
+are defined by utilizing multi-modal information, including
+token ztoken, segment zseg, layout zlay, and visual embed-
+dings zvis as follows:
+z = LN(ztoken + zseg + zlay + zvis) ∈ RL×d,
+where LN is a layer normalization (Ba, Kiros, and Hinton
+2016), and L and d are the length of the input sequence and
+a hidden vector size, respectively. The segment embedding
+indicates which regions are included in the input sequence.
+The layout embedding denotes the encoded bounding box
+coordinates of the token within the image. We normalize all
+coordinates by the size of images and use embedding lay-
+ers to embed x-axis and y-axis features separately. The vi-
+sual embedding is the appearance feature of each region and
+the OCR bounding boxes, which were obtained from Faster-
+RCNN. Note that the layout and visual embeddings are set to
+zero vectors for the task prefix, question, and page number.
+Multi-modal Encoder-Decoder
+Multi-modal encoder.
+Our encoder is a stack of m Trans-
+former blocks, consisting of a self-attention layer and a
+fully-connected layer with residual connections. Following
+FiD (Izacard and Grave 2021), all K input sequences are
+encoded independently and then concatenated to form a uni-
+fied input representation. Formally, we transform each input
+sequence xk into xk ∈ RL×d and concatenate them into
+X ∈ RK×L×d.
+Answer/Arithmetic-expression decoder.
+Our decoder is
+another stack of m Transformer blocks similar to the multi-
+modal encoder, where each block has an additional layer
+of cross-attention between the output sequence and X. The
+answer decoder is modeled as a conditional generation
+pθ(y|X), where θ represents the set of all model parame-
+ters. To allow the model to perform numerical reasoning, we
+train the system to predict annotated arithmetic expressions
+y′ (e.g., “30 − 28”) instead of numeric values y (e.g., “2”)
+
+R
+HIBIGSORby modeling pθ(y′|X). During inference, the model itself
+decides whether numerical reasoning is required or not for
+each question by predicting an indicator token Answer: or
+Expression: at the beginning of the output sequence.
+Evidence selector.
+The selector shares the weights and the
+architecture of the answer/arithmetic-expression decoder.
+Instead of only modeling answer generation, we devise a
+simple method to train evidence selection in a unified se-
+quence. Specifically, we define the output sequence as ˆIpages
+= (Evidence pages: ˆe1, . . ., ˆeK′), where each ˆe is the
+page number of the selected slide.
+Training and inference.
+Our model is trained by mini-
+mizing the weighted sum of two losses L = Ldec + Lsel,
+where Ldec and Lsel are the negative log-likelihood between
+the ground-truth and the prediction regarding the decoder
+and selector, respectively. During inference, we obtain the
+final prediction to post-process the decoded sequence by re-
+moving the task indicator. If an arithmetic expression is gen-
+erated (i.e., Expression: is generated), we use a calcula-
+tor to obtain the final results.
+Experiments
+Experimental Setup
+We conducted experiments on the SlideVQA task, evidence
+selection task, and question answering task respectively de-
+fined in MAINTASK, SUBTASKS 1 and 2.
+Main task baselines.
+We mainly evaluated pipeline mod-
+els as baselines, consisting of evidence selection that pro-
+duces top-3 evidences and question answering that takes the
+selection results as input. Here, we introduced a hierarchical
+LayoutLMv2 (H-LayoutLMv2) inspired by (Tu et al. 2020;
+Xu et al. 2021), which encodes all slides simultaneously by
+using another Transformer layer, as the evidence selector. It
+achieved 96.0% on Recall@3 on the test set. We used three
+generative QA models: a textual model T5 (Raffel et al.
+2020), a numerical and multi-hop model PreasM (Yoran,
+Talmor, and Berant 2022), and a document VQA model
+LayoutT5 (Tanaka, Nishida, and Yoshida 2021). We also
+used an extractive document VQA model LayoutLMv2 to
+predict the single span.
+Evidence selection baselines.
+We also evaluated the ev-
+idence selection task alone. BM25 (Robertson, Zaragoza
+et al. 2009) is a non-neural retrieval framework to estimate
+the relevance of texts to a search query. For the neural mod-
+els, CLIP (Radford et al. 2021) encodes the question and
+each image to predict the highest similar pair. BM25 and
+CLIP used the top-1 slide as the prediction. BERT (Devlin
+et al. 2019) is a pre-trained language model which only uses
+text information with the Transformer architecture. Lay-
+outLM (Xu et al. 2020) incorporates layout information into
+the input embeddings of BERT. LayoutLMv2 includes im-
+age features produced by a CNN backbone in input embed-
+dings. To model the interactions between the slides, we used
+H-LayoutLMv2 described in the previous section. For neu-
+ral evidence selection baselines (except for CLIP), we use a
+hidden state of [CLS] in the last layer to feed into an MLP
+classifier with a sigmoid activation. Evidence is selected if
+its confidence of binary classification is above the optimal
+value on the development set.
+To evaluate the effectiveness of our generative evidence
+selection module, we introduced BinaryClass as a classifi-
+cation baseline, which uses a two-layer MLP classifier with
+a sigmoid activation on top of each encoder representation
+at the start-of-sequence. We also introduced a generative
+baseline, ChainGen, which generates a sequence of selected
+slide page numbers before the answer (Wei et al. 2022).
+Question answering baselines.
+In addition to the pipeline
+models, we developed Q-only, which takes only the ques-
+tion into T5. We also used a VideoQA model UniVL (Luo
+et al. 2020) that can take all of the slide images as input.
+Furthermore, we evaluated our base model FiD (Izacard and
+Grave 2021).
+Human performance.
+We asked six crowdworkers (not
+among those recruited to collect our dataset) to select slide
+images relevant to the question and answer the question.
+Evaluation metrics.
+Following HotpotQA (Yang et al.
+2018), we used exact match (EM) and F1 on each question
+answering and evidence selection task and also used Joint
+EM (JEM) and Joint F1 (JF1) to evaluate both tasks. These
+joint metrics penalize models that perform poorly on either
+task and assess the accuracy and explainability of the ques-
+tion answering models.
+Implementation Details
+We implemented all of the models in PyTorch and experi-
+mented on eight Tesla V100 32GB GPUs. The size of CLIP
+was Large and the size of the other models was Base. We
+fine-tuned the models using AdamW (Loshchilov and Hutter
+2017) with a learning rate of 5e-5 and a dropout rate of 10%,
+and we linearly warmed up the learning rate over 1000 steps.
+The batch size was set to 32. We evaluated models every 500
+steps and selected the best one on the development set on the
+basis of the loss. We used a maximum length of 200 tokens
+for each input sequence of M3D, and set the maximum target
+sequence length to 50. We trained Faster-RCNN (Ren et al.
+2015) with a ResNet-101 (He et al. 2016) backbone by us-
+ing stochastic gradient descent (SGD) (Ruder 2016) with a
+learning rate of 1e-3 and batch size of one. Standard anchor
+scales of [8, 16, 32] and anchor ratios of [0.5, 1.0, 2.0] were
+used. For the VideoQA baseline, we created a new video at
+a rate of five frames per second. We used the Google Cloud
+Vision API to extract text and bounding boxes from images.
+When the OCR word is tokenized into sub-word tokens, the
+bounding box coordinates of a sub-word token are the same
+as those of its whole word.
+Experimental Results and Analysis
+Does our model outperform the baselines?
+Table 2 sum-
+marizes the results of the main tasks. As shown in Table 2a,
+M3D outperformed the baselines on joint EM/F1, where
+the metrics evaluate the consistency between the predicted
+evidence and answers. For the evidence selection task, Ta-
+ble 2b shows that H-LayoutLMv2 and M3D performed bet-
+
+Dev
+Test
+Model
+Modal JEM
+JF1
+JEM
+JF1
+PreasM
+T
+30.2 38.2 23.4 34.7
+T5
+T
+30.0 38.0 22.6 34.2
+T5 + zlay
+TL
+30.9 39.5 23.6 35.7
+LayoutT5
+TLV
+31.7 39.9 24.3 36.1
+LayoutLMv2†
+TLV
+22.8 30.8 16.5 26.5
+M3D
+TLV
+36.2 42.8 28.0 37.3
+M3DGT
+TLV
+44.6 50.4 35.4 44.7
+Human
+–
+–
+–
+88.6 91.9
+(a) Performance of main task.
+Dev
+Test
+Model
+Modal EM
+F1
+EM
+F1
+BM25
+T
+40.1 46.0 35.9 47.5
+CLIPzero
+V
+33.0 34.8 30.6 34.4
+CLIP
+V
+40.6 43.0 39.3 43.5
+BERT
+T
+60.9 74.4 50.3 69.2
+BERT + zlay
+TL
+61.4 75.2 52.7 71.0
+LayoutLM
+TL
+51.0 63.7 42.0 59.9
+LayoutLMv2
+TLV
+63.3 77.1 51.7 71.5
+H-LayoutLMv2
+TLV
+81.1 89.5 69.8 85.6
+M3D
+TLV
+83.1 87.7 75.0 83.8
+Human
+–
+–
+–
+97.7 98.0
+(b) Performance of evidence selection task.
+Dev
+Test
+Model
+Modal EM
+F1
+EM
+F1
+Q-only
+–
+9.4
+11.4 10.7 13.5
+UniVL
+V
+8.8
+12.1 10.6 14.1
+PreasM
+T
+36.3 41.9 30.7 38.2
+T5
+T
+35.2 41.3 29.3 37.9
+T5 + zlay
+TL
+36.9 43.2 31.0 39.7
+LayoutT5
+TLV
+38.9 44.8 31.7 39.9
+LayoutLMv2†
+TLV
+26.5 33.4 21.4 29.3
+FiD
+T
+37.6 42.9 30.4 38.9
+FiD + zlay
+TL
+38.1 43.3 30.6 38.9
+M3D
+TLV
+41.3 47.1 33.5 41.7
+Human
+–
+–
+–
+89.8 93.0
+(c) Performance of question answering task.
+Table 2: Performance of SlideVQA tasks. “T/L/V” denotes the “text/layout/visual” modality of images. †denotes the extractive
+approach. The pipeline models answer the question based on the top-3 evidences obtained by H-LayoutLMv2. M3DGT knows
+the ground-truth evidence. + zlay denotes addition of the layout embedding to the input embeddings. LayoutLM was not pre-
+trained in any matching task (e.g., text-image matching). CLIPzero denotes CLIP without fine-tuning.
+Single-Hop
+Multi-Hop
+Single-Hop &
+Numeric
+Multi-Hop &
+Numeric
+Arithmetic
+Count
+Comparison
+Single-Span
+Multi-Span
+Non-Span
+0
+20
+40
+60
+80
+100
+F1
+FiD
+M3D w/o AE generation
+M3D
+Human
+Figure 6: Performance of models and humans on the answer
+types, reasoning types and numerical operation types in the
+test set. AE stands for “arithmetic expression”.
+ter than the baselines. This indicates that modeling the in-
+teraction between multiple slides simultaneously is needed
+to improve performance. For the QA task, Table 2c shows
+that M3D outperformed the pipeline methods in all met-
+rics. Our end-to-end M3D model is better at ignoring the
+slides irrelevant to the question than the answer generator
+in the pipeline methods that strongly depend on the slides
+narrowed down by the evidence selector. However, M3DGT
+in Table 2a achieved a significant improvement by know-
+ing the ground-truth slides. There is room for improving the
+correctness of evidence selection.
+What are the characteristics of our dataset?
+Table 2
+shows that adding modality information tended to improve
+performance in all tasks. This demonstrates that SlideVQA
+requires methods to have the ability to jointly understand the
+text, layout, and visual modalities of documents. As shown
+in Table 2c, Q-only had the lowest performance, show-
+ing that the systems could not answer the question with-
+out reading documents in the SlideVQA task. Additionally,
+UniVL has a comparative result to Q-only, indicating that
+SlideVQA requires different abilities from VideoQA (Le
+Main
+Select
+QA
+Model
+JEM
+JF1
+EM
+F1
+EM
+F1
+M3D
+36.2
+42.8
+83.1
+87.7
+41.3
+47.1
+w/o AE generation
+35.7
+42.3
+82.9
+87.7
+40.5
+46.3
+w/o Evidence selection
+–
+–
+–
+–
+40.6
+46.4
+w/o Layout features
+35.1
+42.0
+82.4
+87.1
+40.3
+46.3
+w/o Visual features
+34.2
+40.9
+81.5
+86.3
+39.0
+44.9
+w/o Text features
+1.0
+1.5
+8.4
+9.8
+9.8
+12.0
+Table 3: Ablation study of M3D on dev set.
+Main
+Select
+QA
+Model
+JEM
+JF1
+EM
+F1
+EM
+F1
+M3D backbone
+–
+–
+–
+–
+39.0
+44.8
++ BinaryClass
+24.7
+34.8
+54.5
+68.5
+38.8
+44.8
++ ChainGen
+34.0
+40.8
+81.1
+86.1
+39.8
+45.4
++ MultiGen (Ours)
+35.7
+42.3
+82.9
+87.7
+40.5
+46.3
+Table 4: Performance comparison of different evidence se-
+lection methods on dev set.
+and Hoi 2020), especially the ability to read texts in im-
+ages. Tables 2a and 2c show that LayoutT5, a generative
+model, significantly outperformed LayoutLMv2, an extrac-
+tive approach. This result is inline with observations on the
+DROP dataset (Dua et al. 2019), which also has non-span
+answers (Geva, Gupta, and Berant 2020). Additionally, all
+of the models performed all of the tasks significantly worse
+than humans. To be specific, Figure 6 illustrates that (i) bet-
+ter multi-hop reasoning over multiple images is needed and
+(ii) non-span answers to questions involving arithmetic op-
+erations have to be improved.
+Do our sub-modules improve performance?
+Table 3
+lists the results of an ablation study. Here, performance
+consistently decreased as individual modules were removed
+from M3D. This indicates that each of the modules is ef-
+
+Class
+Dev AP
+Test AP
+Title
+86.8
+87.5
+Page-text
+76.9
+76.9
+Obj-text
+29.5
+33.4
+Caption
+25.6
+24.9
+Other-text
+40.5
+39.4
+Image
+60.4
+62.2
+Diagram
+65.4
+64.0
+Figure
+74.1
+68.8
+Table
+67.0
+65.6
+Table 5: Object detection performance of Faster-RCNN
+broken down by bounding box categories. We set an
+intersection-over union (IoU) threshold to 0.5.
+fective. More precisely, the arithmetic expression (AE) gen-
+eration was influential on the QA and Joint performance,
+meaning that predicting the arithmetic expression instead of
+the numerical value enhances the ability to generate answers
+with numerical reasoning. As shown in Figure 6, applying
+AE prediction increased F1 by a large margin (+10.4%) in
+the arithmetic type.
+What are the effective evidence selection methods?
+Ta-
+ble 4 shows that our method, which generates the evidence
+selection and question answering results separately, obtained
+the highest performance. It seems that the generative meth-
+ods (MultiGen and ChainGen) benefited from the text-to-
+text pre-training of T5 more than the classification-based
+method (BinaryClass). Our MultiGen decoder that sepa-
+rately trains evidence selection and question answering had
+the advantage of being easier to train than the ChainGen
+baseline decoder that trains the two tasks as a single se-
+quence generation task.
+On which categories does the object detection model not
+work well?
+Table 5 lists the object detection performance
+of Faster-RCNN broken down by bounding box categories.
+These results show that detecting randomly placed and small
+boxes, such as Obj-text, is more difficult than mostly fixed
+and large boxes, such as Title.
+Qualitative examples.
+Figure 7 demonstrates our model’s
+performance by visualizing a qualitative example. This ex-
+ample needs multi-hop reasoning and an answer involving
+an arithmetic operation. FiD gave an incorrect answer be-
+cause it did not consider the visual layout of the slides.
+Moreover, while LayoutT5 could not understand the process
+of getting numerical answers, M3D successfully extracted
+information (“11%” and “12%”) and generated the same an-
+swer as the ground-truth.
+Discussion and Limitations
+SlideVQA is the largest document VQA benchmark that
+uses multiple images as input and requires multi-hop rea-
+soning; its limitation is that the multi-hop questions created
+by editing are different from the questions humans might ac-
+tually ask the system. We argue that developing models that
+can reason over multiple images is an important research
+direction, and therefore, we employed an editing method
+Copyright ©2014 The Nielsen Company. Confidential and proprietary. 
+8 
+ROCK IS THE BIGGEST GENRE, BUT R&B/HIP-HOP 
+AND POP ARE ALSO STRONG IN 2015 
+30% 
+21% 
+17% 
+9% 
+5% 
+4% 
+3% 
+Rock
+R&B/Hip-Hop
+Pop
+Country
+Latin
+Dance/Elec
+Christian/Gosp
+Share of Total Activity 
+TEA Ratio - 10:1 
+SEA Ratio � 1500:1 
+Copyright ©2014 The Nielsen Company. Confidential and proprietary. 
+9 
+ROCK DOMINATES ALBUMS, POP DRIVES SONG 
+SALES AND R&B/HIP-HOP LEADS STREAMING 
+37% 
+18% 
+12% 
+11% 
+3% 
+2% 
+4% 
+24% 
+23% 
+26% 
+12% 
+2% 
+5% 
+3% 
+23% 
+26% 
+19% 
+5% 
+10% 
+6% 
+3% 
+Rock
+R&B/Hip-Hop
+Pop
+Country
+Latin
+Dance/Elec
+Christian/Gosp
+GENRE SHARE OF TOTAL 
+Album Sales %
+Song Sales %
+Streams %
+Q: What is the combined percentage of Album Sales % and Song Sales % 
+for the genre with a 9% Share of Total Activity?
+GT 
+answer:  23%  
+evidence pages:  8, 9
+FiD
+answer: 57%
+evidence pages: None
+LayoutT5 answer: 68%  
+evidence pages: 8, 9
+M3D 
+answer: 23%  (11% + 12%)
+evidence pages: 8, 9
+p.8
+…
+p.9
+…
+Figure 7: Qualitative example. GT denotes the ground-
+truth. (·) means the generated arithmetic expression. The
+slide deck can be viewed at https://www.slideshare.net/
+musicbizassoc/nielsen-2015-music-biz-presentation-final.
+that guarantees multi-hop questions and easily extends the
+dataset size. Also, our model uses cross-attention on all ev-
+idence candidates, which may cause a computational prob-
+lem when there are a lot of input images (e.g., as in the open-
+domain QA setting like DocCVQA). To remedy this prob-
+lem, we consider that models that train a two-stage selec-
+tor that roughly narrows down candidates to a small number
+of images and then accurately selects evidence images and
+an answer generator in an end-to-end manner are promis-
+ing (Sachan et al. 2021a,b).
+Conclusion
+We introduced a new document VQA dataset, SlideVQA,
+focused on the task of understanding slide decks composed
+of multiple images. We also introduced a unified end-to-
+end model, M3D, that can perform evidence selection and
+question answering tasks and enhance numerical reasoning
+by generating arithmetic expressions. While our evaluation
+highlighted the promise of this approach, it also revealed a
+huge gap compared with human performance, and several
+challenges emerge from multi-hop reasoning on multiple
+images and generating answers with arithmetic operations.
+We believe that our dataset will contribute to the develop-
+ment of intelligent assistant agents that can comprehend di-
+verse real-world documents.
+
+nOEMUS
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+High-Quality Supersampling via Mask-reinforced Deep
+Learning for Real-time Rendering
+Hongliang Yuan1, Boyu Zhang1,2, Mingyan Zhu1,3, Ligang Liu4, Jue Wang1
+1Tencent AI Lab, 2Southeast University, 3Tsinghua University,
+4University of Science and Technology of China
+[email protected], [email protected], [email protected], [email protected], [email protected]
+(a) 0.25-spp input
+(b) NSRR
+(c) RAE
+(d) Ours
+(e) Ground truth
+Figure 1: Left to right: (a) noisy image generated using hybrid path-tracer at 0.25 sample per pixel; (b) Neural supersampling
+network [Xiao et al. 2020] (10.3ms at 1024 × 2048, SSIM: 0.7737); (c) RAE [Chaitanya et al. 2017] (6.5ms, SSIM: 0.7556); (d) our
+sparse sampling reconstruction (7.8ms, SSIM: 0.9036); (e) reference path-traced image with 32768 samples per pixel.
+ABSTRACT
+To generate high quality rendering images for real time applications,
+it is often to trace only a few samples-per-pixel (spp) at a lower res-
+olution and then supersample to the high resolution. Based on the
+observation that the rendered pixels at a low resolution are typically
+highly aliased, we present a novel method for neural supersampling
+based on ray tracing 1/4-spp samples at the high resolution. Our
+key insight is that the ray-traced samples at the target resolution
+are accurate and reliable, which makes the supersampling an inter-
+polation problem. We present a mask-reinforced neural network
+to reconstruct and interpolate high-quality image sequences. First,
+a novel temporal accumulation network is introduced to compute
+the correlation between current and previous features to signifi-
+cantly improve their temporal stability. Then a reconstruct network
+based on a multi-scale U-Net with skip connections is adopted for
+reconstruction and generation of the desired high-resolution image.
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from [email protected].
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+© 2018 Association for Computing Machinery.
+ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00
+https://doi.org/XXXXXXX.XXXXXXX
+Experimental results and comparisons have shown that our pro-
+posed method can generate higher quality results of supersampling,
+without increasing the total number of ray-tracing samples, over
+current state-of-the-art methods.
+KEYWORDS
+Monte Carlo denoising, neural networks, path tracing
+1
+INTRODUCTION
+Rendering noise-free Monte Carlo (MC) ray-traced images at real-time
+frame rates is still challenging. Despite the widely used of modern RTX
+GPU accelerators, only a few rays per pixel can be traced at target resolution
+for real-time applications, resulting in severe noise in renderings. The most
+efficient strategy is to denoise and reconstruct the rendering results in
+image-space, usually as a post-process pass of a physically-based renderer.
+Until recently, most MC denoisers were proposed based on convolutional
+neural networks (CNN). Chaitanya et al. [Chaitanya et al. 2017] proposed
+a recurrent model for interactive applications that are targeted at images
+rendered with low sample per pixel (1~4 spp). In addition, the NVIDIA
+OptiX ray-tracing engine introduces an AI-accelerated denoiser based on
+this work. We also developed a hybrid ray tracer based on Vulkan and we
+use it to export training datasets. The source code of our ray tracer and
+paper will be available soon. Mustafa et al. [Işık et al. 2021] adopt dilated
+arXiv:2301.01036v1  [cs.CV]  3 Jan 2023
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+spatial kernels to filter the noisy image-guided by pairwise affinity over
+the features and target in the low-sample count regime (2~8 spp). Meng et
+al. [Meng et al. 2020] also denoise 1-spp noisy input images with a neural
+bilateral grid at real-time frame rates. Hasselgren et al. [Hasselgren et al.
+2020] proposed a neural temporal adaptive sampling method for denoising
+image sequences rendered at 4-spp. Fan et al. [Fan et al. 2021] expands the
+kernel-prediction method to remove noise at low spp (more than one) in
+a strict time budget. All the state-of-the-art denoising and reconstruction
+methods aim at removing noise of images rendered with more than 1-spp.
+In this paper, we propose a novel approach to reconstruct less than 1-spp
+renderings at real-time frame rates. Following traditional temporal anti-
+aliasing [Karis 2014] (TAA), our method uses renderer generated motion
+vector to warp previous frames and accumulate sparse samples from the pre-
+vious frame based on the temporal accumulation factor computed according
+to the correlation of current and previous frame, effectively increasing the
+number of samples per pixel. The module can also detect ghosting arti-
+facts at disocclusion regions and remove mismatched pixels at inconsistent
+shading regions. Mustafa et al. [Işık et al. 2021] also compute temporal
+accumulation factor for a pixel using neural network, but they concatenate
+features of current and previous frames and feed them into network to-
+gether. Compared to this method, our method can produce better temporal
+stable and high-quality results.
+After accumulating sparse samples, we use a residual block [He et al. 2015]
+to fusion the accumulated features. Then we implement a multi-scale U-Net
+[Ronneberger et al. 2015a] with skip connections for the reconstruction
+subnetwork. The multi-scale predicting network is similar to the method
+suggested by Vogels et al. [Vogels et al. 2018] which uses kernel prediction.
+We directly predict denoised images for the current frame and two additional
+channels as blending factors. We also predict a 2 × downscaled image from
+the layer of the last but one. We composite the final denoised image from the
+current denoised image, the previous warped image, and 2 × downscaled
+images. Comprehensive experiment results show that our approach is good
+at reconstructing 0.25-spp images at a real-time frame rate. To summarize,
+our contributions are the following:
+• We introduce a temporally-stable neural network to reconstruct image
+sequences rendered at 0.25-spp at real-time frame rates. To the best of
+our knowledge, we are the first that utilize 0.25-spp images as input
+for the neural network.
+• A novel temporal accumulation network which computes the correla-
+tion between current and previous features to significantly improve
+the temporal stability of Monte Carlo denoising.
+• Extensive experiments demonstrate that our method outperforms state-
+of-the-art methods both quantitatively and qualitatively.
+2
+RELATED WORK
+Traditional best-performing MC denoisers were mainly based on local neigh-
+borhood regression models [Zwicker et al. 2015]. With the advent of power-
+ful modern GPUs, lots of researchers utilize CNN to build their MC denoisers.
+In this section, we will mainly discuss CNN-based real-time denoising tech-
+niques, which are most related to our approach. For a comprehensive study
+of deep learning-based MC denoising and reconstruction techniques, please
+refer to the recent survey of Huo et al. [Huo and Yoon 2021].
+2.1
+Image-space Methods
+Traditional MC denoisers are based on zero-order regression [Delbracio et al.
+2014; Kalantari et al. 2015; Li et al. 2012; Moon et al. 2013; Rousselle et al.
+2012, 2013], first-order regression [Bauszat et al. 2011; Bitterli et al. 2016;
+Moon et al. 2014] and even higher-order regression models [Moon et al.
+2016].The filtering-based methods are based on using the auxiliary feature
+buffers to guide the construction of image-space filters. Most of the above
+methods run in offline rendering. To increase the effective sample count,
+real-time denoisers leverage temporal accumulation between frames over
+time to amortize supersampling [Yang et al. 2009], i.e. temporal anti-aliasing
+(TAA). The previous frame is reprojected according to the motion vector
+and blended with the current frame using a temporal accumulation factor 𝛼.
+The 𝛼 can be constant [Mara et al. 2017; Meng et al. 2020; Schied et al. 2017]
+and changed [Schied et al. 2018] per frame and per pixel. The fixed temporal
+accumulation factor inevitably leads to ghosting and temporal lag. By setting
+the parameter adaptively, the temporal filter can fastly respond to times
+in case of sudden changes between frames. Yang et al. [Yang et al. 2020]
+survey recent TAA techniques and provide an in-depth analysis of the image
+quality trade-offs with these heuristics. Koskela et al. [Koskela et al. 2019]
+propose a blockwise regression for real-time path tracing reconstruction
+and also do accumulation to improve temporal stability.
+2.2
+CNN-based Monte Carlo Denoising
+Recent deep learning denoisers [Bako et al. 2017; Vogels et al. 2018] use
+deep CNN to estimate the local per-pixel filtering kernels used to compute
+each denoised pixel from its neighbors. Dahlberg et al. [Dahlberg et al. 2019]
+implement the approach of [Vogels et al. 2018] as a practical production
+tool used on the animated feature film. Layer-based denoiser [Munkberg
+and Hasselgren 2020] designs a hierarchical kernel prediction for multi-
+resolution denoising and reconstruction. Since the high computational cost
+of predicting large filtering kernels, these methods mostly target offline
+renderings. There are also other methods [Gharbi et al. 2019; Kuznetsov
+et al. 2018; Xu et al. 2019; Yu et al. 2021] that target denoising rendering
+results at more than 4 spp.
+To reduce the kernel prediction methods’ overhead, Fan et al. [Fan et al.
+2021] predict an encoding of the kernel map, followed by a high-efficiency
+decoder to construct the complete kernel map. Chaitanya et al. [Chaitanya
+et al. 2017] proposed a recurrent connection based on U-Net [Ronneberger
+et al. 2015b] to improve temporal stability for sequences of sparsely sampled
+input images. Hasselgren et al. [Hasselgren et al. 2020] proposed a neural
+spatio-temporal joint optimization of adaptive sampling and denoising with
+a recurrent feedback loop. Hofmann et al. [Hofmann et al. 2021] also utilized
+the neural temporal adaptive sampling architecture to denoise rendering
+results with participating media. Xiao et al. [Xiao et al. 2020] presented a
+neural supersampling method for TAA, which is similar to deep-learned
+supersampling (DLSS) [Edelsten et al. 2019]. Meng et al. [Meng et al. 2020]
+denoised 1-spp noisy input images with a neural bilateral grid at real-time
+frame rates. Mustafa et al. [Işık et al. 2021] adopted dilated spatial kernels
+to filter the noisy image guiding by pairwise affinity over the features.
+Compare with these real-time denoising framework targeting for more than
+1-spp renderings, our method is designed to work with 0.25-spp.
+3
+SPARSE SAMPLING DENOISING
+3.1
+Problem Statement
+Our goal is to reconstruct temporally stable video from 0.25-spp hybrid path
+traced image sequences in real-time frame rates, and we achieve this with a
+supervised deep learning method. We use our hybrid path traced renderer to
+generate a set of data D={(c1,f1,r1), ... ,(c𝑁 ,f𝑁 ,r𝑁 )} where c stands for noisy
+image rendered by sparse sampling, f is the auxiliary features (e.g. albedo,
+normal, depth, metallic, roughness, shadow and transparent) obtained in the
+rendering process, r is the reference image with high spp. We train a deep
+neural function Φ with parameters Θ to reconstruct the noisy-free image.
+The loss function ℓ is measured as the difference between the denoised
+image and its reference image. We then minimize the loss function with
+gradient descent algorithm across the dataset D with 𝑁 samples to get the
+
+High-Quality Supersampling via Mask-reinforced Deep Learning for Real-time Rendering
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+optimal parameters ˆ𝜃:
+ˆ𝜃 = arg min
+𝜃
+𝑁
+∑︁
+𝑖=1
+ℓ (Φ(𝑐𝑖, f𝑖),𝑟𝑖)
+(1)
+The loss function combines four-loss items, including spatial, temporal,
+relative edge, and albedo loss, see section 3.4 for details.
+3.2
+Sparse Sampling
+We developed a hybrid ray tracer to generate our dataset. To accelerate ray
+tracing, we leverage a rasterization pipeline to get the first hit position from
+the camera and store its associated shading attributes including albedo, nor-
+mal, depth, motion vector, metallic, and roughness. After this rasterization
+pass, we trace a shadow ray to record the soft shadow attribute. If there are
+transparent materials in the scene, we also save the transparent attribute
+at the first hit position. We divide the full resolution into non-overlapping
+blocks with spatial size 4 × 4. We use the MC method to solve the rendering
+equation [Kajiya 1986] for one pixel in the block at each frame and other
+pixels remain zero, see Figure 2. If the camera is static, the radiance of all
+pixels will be computed once at every four frames. For image sequences,
+we use two-layer CNNs to accumulate history frames, see section 3.3.1.
+t
+t+1
+t+2
+t+3
+Figure 2: Sampling pattern. In t and t+1 frame, we compute
+radiance for the top left and top right pixel, respectively. In
+t+2 and t+3 frame, bottom left and right pixel is estimated,
+respectively.
+The input for our network is 18-channels features, including a 3D vector
+(noised image, albedo, normal, shadow, and transparent) and a 1D vector
+(depth, metallic, and roughness). Following prior method [Chaitanya et al.
+2017], we demodulate the noisy RGB image by the albedo of the directly
+visible material, and the untextured irradiance 𝑥 is transformed to log space,
+ln(1 + 𝑥). Different from the prior method [Chaitanya et al. 2017], after
+the untextured irradiance has been reconstructed, we re-modulate by the
+accumulated albedo predicted by our temporal accumulator network which
+is our key module for producing temporally stable results.
+3.3
+Network Pipeline
+In this section, we describe our method in details with Figure 3.
+3.3.1
+Temporal Accumulator. The temporal accumulator module contains
+two neural networks each with 2-layer CNNs. One network accepts normal
+and depth of current frame as input and outputs reference embedding.
+Another network computes embeddings for the current frame and warped
+the previous frame. These two embeddings are then multiplied in a pixel-
+wise manner to the reference embedding and then call softmax(·) to get 𝛼
+and 𝛽 (𝛼 +𝛽 = 1) blending factors for current features and previous features,
+respectively. We only accumulate noisy images, shadow and albedo (see
+Figure 4). Take shadow as an example,we use the following equation to
+accumulate shadow over the frame:
+f𝑠
+𝑡 = 𝛼 W(f𝑠
+𝑡−1) + 𝛽f𝑠
+(2)
+where f𝑠
+𝑡 is accumulated shadow until 𝑡 frame, f𝑠 is shadow buffer for 𝑡
+frame. W(·) is a warping operator that reprojects previous frame to current
+one using motion vector. For the first frame, we set f𝑠
+𝑡−1 to f𝑠.
+3.3.2
+Feature Fusion. After accumulating images, shadow, and albedo, we
+concatenate accumulated features, normal, depth, transparent, metallic, and
+roughness. Then we feed them into a feature fusion network. Since our
+image is sparse, we use this network to fusion the features and spread
+signals across spatial space.
+3.3.3
+Reconstruction Network. Finally, fused features and warped denoised
+images of the previous frame are concatenated and fed into a reconstruc-
+tion network, which outputs the high-quality image for the current frame.
+The reconstruction network details are given in Figure 3. Our network di-
+rectly predict denoised fine image d𝑓 for current frame and two additional
+channels as blending factor, i.e., 𝛼𝑠 and 𝛼𝑡. We also directly predict a 2 ×
+downscaled coarse image d𝑐 from the layer of the last but one. We use scale
+composition suggested by Vogels et al [Vogels et al. 2018] to combine fine
+and coarse images:
+O𝑝 = d𝑓
+𝑝 − 𝛼𝑠
+𝑝 [UDd𝑓 ]𝑝 + 𝛼𝑠
+𝑝 [Ud𝑐 ]𝑝
+(3)
+where D and U are 2 × 2-downsampling and nearest-neighbor upsampling
+operators. The filtered history O𝑡−1 is linearly blended with the result of
+the scale composition O using 𝛼𝑡:
+O𝑡 = 𝛼𝑡O + (1.0 − 𝛼𝑡)O𝑡−1
+(4)
+3.4
+Losses
+We use the symmetric mean absolute percentage error (SMAPE):
+ℓ (r, d) =
+1
+3𝑁
+𝑝=𝑁
+∑︁
+𝑝=1
+𝑐=3
+∑︁
+𝑐=1
+��d𝑝,𝑐 − r𝑝,𝑐
+��
+��d𝑝,𝑐
+�� +
+��r𝑝,𝑐
+�� + 𝜀
+(5)
+Here, 𝑁 is the number of pixels in image and 𝜀 is 10−2. d and r are the
+denoised frame and the corresponding reference frame.
+Our loss combines two parts, the first one is computed on a sequence of
+5 images, including spatial loss ℓ𝑠 = ℓ (r, d), temporal loss ℓ𝑡 = ℓ (Δr, Δd)
+where Δ is temporal gradient computed between two consecutive frames,
+relative edge loss ℓ𝑒 = 𝐿1( ∇d
+r+𝜀 , ∇r
+r+𝜀 ), where gradient ∇ is computed using
+a High Frequency Error Norm (HFEN), an image comparison metric from
+medical imaging [Ravishankar and Bresler 2011]. As suggested by Chaitanya
+et al. [Chaitanya et al. 2017], we assign higher weight to three loss functions
+(ℓ𝑠, ℓ𝑡 and ℓ𝑒) of frames later in the sequence to amplify temporal gradients.
+For our training sequence of 5 images, we use (0.05, 0.25, 0.5, 0.75, 1).
+The second part is warped temporal loss ℓ𝑤𝑡 = ℓ (𝜔r,𝜔d) where 𝜔r =
+𝑟4 − W(𝑟3), W(·) is a warping operator that reprojects previous frame
+to current one. We also include albedo loss ℓ𝑎 = ℓ (a𝑎𝑐𝑐, a𝑟 ) where a𝑎𝑐𝑐 is
+accumulated albedo computed by our feature accumulator network. We
+only compute albedo loss on last frame and warped temporal loss on last
+two frames.
+We use a weighted combination of these losses as the final training loss:
+ℓ = 0.7ℓ𝑠 + 0.1ℓ𝑡 + 0.2ℓ𝑒 + 0.4ℓ𝑤𝑡 + 5.0ℓ𝑎
+(6)
+4
+DATESET AND TRAINING PROCEDURE
+Since our method is designed for 3A game and virtual character rendering,
+we train a separate network for each 3D scene same as [Xiao et al. 2020].
+Due to the input image being generated at 0.25-spp, training robust denoiser
+requires a large number of images. We train our method on 6 scenes, see
+Figure 5). BistroInterior and BistroExterior [Lumberyard 2017] have more
+than one million triangles and support transparency, diffuse, specular, and
+soft shadow features. Sponza, Diningroom, Angel, and Warmroom are
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Reconstruction  Network 
+Reconstruction  Network 
+Feature Fusion
+Feature Accumulator
+ReLu
+Conv+ReLu
+Current Features  
+�
+�
+32
+32
+32
+32
+Conv+ReLu
+Conv+ReLu
+Reconstructed 
+Image
+Conv+ReLu
+·
+·
+Reference
+Warp
+43
+C
+Conv+ReLu
+Conv
+Conv+ReLu
+Conv
+43
+43
+32
+Downsample
+Conv+ReLu
+Conv+ReLu
+48
+48
+Downscale
+Conv+ReLu
+Conv+ReLu
+64
+64
+Downscale
+Conv+ReLu
+Conv+ReLu
+64
+64
+Downscale
+Conv+ReLu
+Conv+ReLu
+80
+80
+Downscale
+Conv+ReLu
+Conv+ReLu
+80
+80
+Downscale
+Conv+ReLu
+Conv+ReLu
+96
+96
+Downscale
+Conv+ReLu
+Conv+ReLu
+96
+96
+Conv
+ReLu
+Conv
+ReLu
+128
+128
+Conv
+ReLu
+Conv
+ReLu
+128
+128
+Upsample
+Conv+ReLu
+Conv+ReLu
+96
+80
+Upsample
+Conv+ReLu
+Conv+ReLu
+96
+80
+Upsample
+Conv+ReLu
+Conv+ReLu
+Upsample
+Conv+ReLu
+Conv+ReLu
+96
+64
+Upsample
+Conv+ReLu
+Conv+ReLu
+96
+64
+Upsample
+Conv+ReLu
+Conv+ReLu
+64
+48
+Upsample
+Conv+ReLu
+Conv+ReLu
+48
+32
+Upsample
+Conv+ReLu
+Conv+ReLu
+48
+32
+Conv+ReLu
+Upsample
+Conv+ReLu
+64
+5
+Softmax
+64
+32
+Conv+ReLu
+Downsample
+Conv+ReLu
+Conv+ReLu
+Downsample
+Conv+ReLu
+Conv
+3
+Previous out
+Warp
+Coarse 
+ReLu
+Composition
+ReLu
+Upsample
+2x
+2x
+Upsample
+2x
+Downscale
+2x
+2x
+Downscale
+2x
+Normal&Depth
+Normal&Depth
+f tf t
+f
+1
+-
+tf
+1
+-
+t
+Previous Features
+Previous Features
+O
+1
+-
+t
+O
+1
+-
+t
+Ot
+Ot
+Figure 3: Network pipeline of our sparse sampling reconstruction (SSR) method. The pipeline includes feature accumulator,
+feature fusion, and reconstruction networks. The numbers under each network layer represent the output channels at cor-
+responding layers. The kernel size is 3 × 3 at all layers. The operator ⊙ denotes dot product between features. c○ indicates
+concatenation operation. ⊕ and ⊗ represent element-wise addition and multiplication, respectively.
+(a) warped albedo
+(b) current albedo
+(c) accumulated albedo
+Figure 4: The history albedo (a) is first wared and then is
+blended with the current frame albedo (b). Our temporal ac-
+cumulator not only fills missing pixels but also smooths ar-
+tifacts at the edge.
+simple scenes. Each scene in the training set contains 100 to 1000 frames
+with resolution 1024 × 2048 depending on its complexity. We also rendered
+a validation set with 10 frames and a test set with 50 frames for each scene.
+For each frame, we rendered the reference image at 32768 spp which is the
+target of our denoiser.
+(a) BistroInterior
+(b) BistroExterior
+(c) Sponza
+(d) Diningroom
+(e) Angel
+(f) Warmroom
+Figure 5: An overview of reference images in our generated
+dataset
+When we train the denoiser, we randomly select 5 consecutive frames for
+training in consecutive clips of each scene. The inputs, including the noisy
+image and the auxiliary features, of each frame, are randomly cropped with
+resolution 256 × 256 to make full use of GPU.
+We optimize our denoiser network with ADAM optimizer [Kingma and Ba
+2015]. We set the initial learning rate to 1 × 10−4 and half it at one-third
+and two-thirds of the total number of iterations. The batch size is 7 and
+the epoch is 200 for each scene. Our denoiser is implemented by PyTorch
+[Paszke et al. 2019] and all the models we presented were trained and tested
+parallel on four GPUs of NVIDIA Tesla A100. Each network takes around 9
+hours.
+5
+RESULTS
+In this section, we evaluate the performance of our method. We describe the
+implementation of compared baseline and metrics in Section 5.1, analyze
+the algorithm with various ablation experiments in Section 5.2, and describe
+its limitations and future work in Section 5.3.
+5.1
+Baseline and metrics
+We compare our method with several state-of-the-art denoising and recon-
+struction work, including real-time methods RAE [Chaitanya et al. 2017],
+ANF [Işık et al. 2021], offline method MCD [Yu et al. 2021], and super-
+resolution model NSRR [Xiao et al. 2020]. Although NSRR is a method for
+the super-resolution task, it can also reconstruct images from zero-padding
+inputs which means it fits the sparse sampling task well. So we choose it
+as one of our competitors. We removed the zero-sampling modules so that
+it can apply to our dataset. We follow all these papers and use PyTorch to
+re-implement them. We train all the methods on the same datasets as in our
+method with the same training procedure.
+To evaluate quality, we use three quality metrics: peak signal to noise ratio
+(PSNR), structural similarity index (SSIM) [Wang et al. 2004], and root mean
+squared error (RMSE). The higher the better in both PSNR and SSIM, while
+the lower the better in RMSE.
+Quantitative comparison results are shown in Table1. Average results are
+reported on the 50 test videos of six scenes. We only show the results of
+SSIM due to space limit, and please refer to our supplemental material for
+more comparison results. As shown in Table 1, our method achieves the
+best performance with all six scenes.
+At inference time, all methods are applied to a single frame at a time, Table
+2 shows inference time at 1024 × 2048 resolution. We tested all the models
+using a GPU, NVIDIA Tesla A100. All network models are not optimized
+with Nvidia TensorRT at 16-bit precision, so inference time still has room
+for improvement.
+
+ERCESTTTHigh-Quality Supersampling via Mask-reinforced Deep Learning for Real-time Rendering
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+Scene
+MCD
+ANF
+NSRR
+RAE
+SSR
+BistroInterior
+0.7650
+0.7583
+0.7405
+0.7751
+0.8921
+BistroExterior
+0.8071
+0.7201
+0.8538
+0.8006
+0.8962
+Sponza
+0.8119
+0.8219
+0.8113
+0.8898
+0.9410
+Diningroom
+0.8637
+0.7226
+0.8843
+0.9007
+0.9375
+Warmroom
+0.8021
+0.8774
+0.9740
+0.9675
+0.9758
+Angel
+0.8601
+0.8813
+0.9804
+0.9161
+0.9763
+Table 1: Quantitative comparison results on six scenes. We
+choose four baseline methods to compare with our SSR
+method.
+Method
+MCD
+ANF
+NSRR
+RAE
+SSR
+Time(ms)
+13.5
+32
+33.5
+6.5
+7.8
+Table 2: Comparison results of inference time.
+In Figure 6, we compare reconstructed images visually. Our method out-
+performs all other methods on all scenes by a large margin. Previous state-
+of-the-art methods are not good at denoising renderings at 0.25-spp. MCD
+originally targets offline rendering and transformer needs large memory to
+train and inference. RAE, NSRR, and ANF feed previous and current features
+into the network directly. The difference between our approach and the
+previous ones is that we compute the correlation for each pixel between
+normal and depth features of current and previous frame. Please refer to
+the supplementary material and videos, our method produces significantly
+more temporally stable video results than existing methods.
+5.2
+Analysis
+5.2.1
+Rendering Efficiency. We test rendering time of each stage in NVIDIA
+RTX 3060 GPU at resolution 1024 × 2048, see Table 3. With our sparse
+sampling, the total rendering time of scene BistroInterior is 8.75 ms. Without
+sparse sampling, the total rendering time is 18.19 ms. This leads to an about
+3× rendering performance improvement. After applying our SSR model,
+high-fidelity results are produced.
+Rasterization
+Transparent and Shadow
+W-SS
+Wo-SS
+1.08 ms
+2.32 ms
+5.35 ms
+14.79 ms
+Table 3: Rendering time of scene BistroInterior. W-SS means
+rendering with our sparse sampling, Wo-SS means without
+sparse sampling.
+5.2.2
+Quality Gain with Shadow and Transparent. Our training images are
+produced by MC path tracer at 0.25-spp average. Due to light occlusion,
+more than three-fourths of pixels remain zero, so we need more features to
+train our model. We add direct noisy shadow as input of our model. Our
+feature accumulator will accumulate noisy shadows between the current
+frame and the history shadow buffer. The accumulated shadow can help
+our model to detect the continuous edge of shadow and improve temporal
+stability. The synthesis video of test sequences can show that the edge of
+Figure 7 without shadow feature will jitter over frames. If noisy features
+feed into regression-based method [Rousselle et al. 2013], the quality of the
+denoised image will decrease. These methods need another filter to prefilter
+noisy features, but CNN-based methods can accept more than one noisy
+buffer except noisy images.
+We also add the transparent feature into our model for training, but we
+did not accumulate it before feeding it into the feature fusion module. The
+reason is that the transparent feature includes less noise than the shadow, see
+Figure 8. If the scene didn’t have a transparent object, such as BistroExterior,
+we also feed transparent features with zero.
+Our model without shadow and transparent only gets 27.98 dB on testing
+BistroInterior. With shadow and transparent, our model not only gets higher
+PSNR (28.78 dB) but also generates high-quality image.
+5.2.3
+Quality Gain with Feature Accumulator. We demodulate the image
+with the albedo at a primary hit position. After our network reconstructs
+the untextured illumination, we re-modulate by the albedo to include the
+texture detail in the final rendering. If the albedo in the corresponding frame
+has an artifact, the artifact will transfer to the final rendering. Chaitanya et.
+al [Chaitanya et al. 2017] apply TAA as a supplemental post-process pass to
+fix the artifact. We re-modulate by the accumulated albedo generated by the
+temporal accumulator module to achieve some efficiency as multisampling
+antialiasing (MSAA). [Akeley 1993].
+In summary, from Figure 4, Figure 7 and Figure 9, we can see that our feature
+accumulator plays a key role in reconstructing sparse sampling renderings
+at less than 1-spp.
+5.2.4
+Network Modules. In Table 4, we report the ablation experiments
+for analyzing the quality improvements from the temporal accumulator
+(Section 3.3.1) and feature fusion (Section 3.3.2) modules. Average results
+are reported on the 50 test image sequences of the BistroInterior scene. If
+without the temporal accumulator and feature fusion, the PSNR decreases
+about 0.58dB, but temporal stability will decrease dramatically in the video
+results. See our supplemental materials for more detailed information.
+Feature Accumulator
+Feature Fusion
+SSIM
+PSNR (dB)
+�
+�
+0.8600
+28.20
+�
+�
+0.8617
+28.15
+�
+�
+0.8866
+28.56
+�
+�
+0.8911
+28.78
+Table 4: Ablation experiment for the feature accumulator
+and feature fusion modules. The network is trained with
+each (and both) of these subnetworks removed, and results
+on the BistroInterior scene are reported.
+5.3
+Limitations and Future Work
+While our method provides a significant improvement for neural sparse
+sampling reconstruction, the inference time still has room for improvement.
+We will adopt TensorRT for acceleration and deploy our model on our game
+engine and virtual character rendering platform in the future. In addition,
+there is still little jitter for small objects on the temporal domain. Modern
+game engine all has a TAA pass, applying TAA post-processing can get more
+temporally stable results. We also try to add a layer of Swin Transformer
+[Liu et al. 2021] to the first layer of our reconstruction network. It can truely
+improve quantitative number about 0.23 dB, but inference time will increase
+1.1 ms at 1024 × 2048 resolution on NVIDIA Tesla A100.
+6
+CONCLUSION
+We have presented the first CNN-based method for reconstructing Monte
+Carlo renderings at 0.25-spp and experiments show that our method recon-
+structs high-quality results compared with current state-of-the-art methods.
+
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+(a) Ours
+(b) Input
+(c) MCD
+(d) ANF
+(e) NSRR
+(f) RAE
+(g) Ours
+(h) Reference
+Figure 6: Visual results on BistroInterior, BistroExterior, Sponza, Diningroom, Warmroom, and Angel scenes.
+
+ALLLAHigh-Quality Supersampling via Mask-reinforced Deep Learning for Real-time Rendering
+Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
+(a) Wo-shadow
+(b) Ours
+(c) Noisy shadow
+(d) Ground truth
+Figure 7: The result (a) is generated by training model with-
+out shadow feature, our result (b) is trained with shadow fea-
+ture. (c) and (d) is noisy shadow feature and ground truth,
+respectively.
+(a) Wo-transparent (b) W-transparent
+(c) Transparent
+(d) Ground truth
+Figure 8: The result (a) is generated by training our model
+without the transparent feature, our result (b) is trained with
+the transparent feature. (c) and (d) is the transparent feature
+and ground truth, respectively.
+(a) SSIM: 0.8626
+(b) SSIM: 0.8972
+(c) Ground truth
+Figure 9: (a) Artifact is transferred to the final result, SSIM
+is 0.8626 (b) Re-modulating by the accumulated albedo leads
+to high-quality image, SSIM is 0.8927.
+We propose an efficient feature accumulator network to compute the blend-
+ing factor for each pixel between current and previous frames. Then the
+accumulated features are fused and fed into a multi-scale U-Net to recon-
+struct final results. We evaluated our method by comparing its performance
+to previous works demonstrating better results across all test scenes.
+7
+ACKNOWLEDGMENTS
+We thank Open Research Content Archive (ORCA) of NVIDIA for provid-
+ing BistroInterior and BistroExterior scenes for training and testing. We
+also thank all students who participate in the development of our hybrid
+renderer.
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