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+arXiv:2301.03732v1  [math.DG]  10 Jan 2023
+A SCHUR’S THEOREM VIA A MONOTONICITY AND THE
+EXPANSION MODULE
+LEI NI
+Abstract. In this paper we present a monotonicity which extends a classical theorem
+of A. Schur comparing the chord length of a convex plane curve with a space curve of
+smaller curvature. We also prove a Schur’s Theorem for spherical curves, which extends
+the Cauchy’s Arm Lemma.
+1. Introduction
+For a convex curve c(s) : [0, L] → R2 and a smooth curve in ˜c(s) : [0, L] → R3 of the same
+length (both parametrized by the arc-length), A. Schur’s theorem [7] (Theorem A page 31,
+see also [5]) asserts that if both curves are embedded, and the curvature of the space curve
+˜k(s) := | ˜T ′|(s), where ˜T(s) = ˜c′(s) is the tangent vector, is not greater than the curvature
+k(s) of the convex curve, then d(˜c(0), ˜c(L)) ≥ d(c(0), c(L)). From the proof of [7] it is easy
+to see R3 can be replaced by Rn with n ≥ 2.
+The theorem can be proven for curves whose tangents have finite discontinuous jumps,
+and to the situation that the curvature of the smaller curve is a curve in Rn+1 for n ≥ 1.
+In terms of the generalization to curves with finite discontinuous points for the tangent, it
+assumes that there exists {sj}0≤j≤N such that 0 = s0 < s1 < · · · < sk < · · · < sN = L such
+that both c(s) and ˜c(s) are regular embedded curves for s ∈ (sj−1, sj) for all 1 ≤ j ≤ N
+satisfying k(s) ≥ ˜k(s), and for each 1 ≤ j ≤ N − 1 at the point c(sj) and ˜c(sj), the oriented
+turning angles, which are measured by signed distance αj := dSn(c′(sj−), c′(sj+)) > 0 and
+˜αj = dSn(˜c′(sj−), ˜c′(sj+)), satisfy that αj ≥ ˜αj for all 1 ≤ j ≤ N − 1. The convexity of
+c(s) and the simpleness assumption imply that
+N
+�
+j=1
+� sj
+sj−1
+k(s) ds +
+N−1
+�
+j=1
+αj ≤ 2π.
+(1.1)
+This extension, together with some ingenious applications of the hinge’s theorem, allows
+one to prove the famous Cauchy’s Arm Lemma for geodesic arms in the unit sphere (consist-
+ing of continuous broken great/geodesic arcs with finite jumps of the tangents) in Lemma
+II on the pages 37–38 of [7]. The Lemma became famous due to that it had an incom-
+plete/false proof by Cauchy originally [4]. The corrected proof appeared in [1, 11]. This
+spherical Cauchy’s Arm Lemma can also be proved by an induction argument [12], whose
+idea in fact in part resembles the proof of the smooth case to some degree. Note that this
+lemma of Cauchy plays a crucial role in the rigidity of convex polyhedra in R3, which finally
+was vastly generalized to convex surfaces (convex bodies enclosed) as the famous Pogorelov
+monotypy theorem (cf. [3] Section 21).
+1
+
+2
+LEI NI
+The Schur’s theorem also can be applied to prove the four-vertex theorem for convex plane
+curves, besides implying a Theorem of H. A. Schwartz which asserts: For any curve c of
+length L with curvature k(s) ≤ 1/r, let C be the circle passing c(0) and c(L) of radius r,
+then L is either not greater than the length of the lesser circular arc, or not less than the
+length of the greater circular arc of C.
+High dimensional (intrinsic) analogues of A. Schur’s
+theorem include the Rauch’s comparison theorem and the Toponogov comparison theorem.
+The later however has the limit of requiring that the manifold with less curvature must be
+a space form of constant sectional curvature.
+First we have the following slight more general version of Schur’s theorem in terms of a
+monotonicity.
+Theorem 1.1. Let c : [0, L] → R2 be an embedded convex plane curve with curvature
+k(s) ≥ 0. Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s). Then for any
+0 ≤ s′ < s′′ ≤ L there exist an isometric inclusion ιs′,s′′ : R2 → Rn+1 with ιs′,s′′(0) = 0
+such that
+I(s) := ⟨˜c(s) − ιs′,s′′(c(s)), ιs′,s′′(c(s′′) − c(s′))⟩
+is monotone non-decreasing for s ∈ [s′, s′′], or equivalently
+⟨ ˜T(s) − ιs′,s′′(T (s)), ιs′,s′′(c(s′′) − c(s′))⟩ ≥ 0,
+∀ s ∈ [s′, s′′].
+(1.2)
+As s′ → s′′, the inclusion ιs′,s′′ converges to an inclusion identifying T (s) with ˜T(s).
+Corollary 1.2. Under the same assumption as in the theorem, for any s′ ≤ s′
+∗ < s′′
+∗ ≤ s′′,
+⟨c(s′′
+∗) − c(s′
+∗), c(s′′) − c(s′)⟩ ≤ ⟨˜c(s′′
+∗) − ˜c(s′
+∗), ιs′,s′′(c(s′′) − c(s′))⟩.
+(1.3)
+When s′ = s′
+∗ and s′′ = s′′
+∗ we have that
+|c(s′′) − c(s′)|2 ≤ ⟨˜c(s′′) − ˜c(s′), ιs′,s′′(c(s′′) − c(s′))⟩.
+(1.4)
+The estimate (1.4) implies Schur’s theorem by the Cauchy-Schwarz inequality applied to
+the right hand side of (1.4):
+|c(s′′) − c(s′)| ≤ |˜c(s′′) − ˜c(s′)|,
+∀ 0 ≤ s′ < s′′ ≤ L.
+This extension allows one to rephrase the result in terms of the concept of the expansion
+module [2, 9] of vector fields. If X : Ω ⊂ Rn+1 → Rn+1 is a vector field defined on a convex
+domain, then the expansion module is a function of one variable ψ(t) such that
+⟨X(y) − X(x), y − x
+|y − x|⟩ ≥ 2ψ
+�|x − y|
+2
+�
+.
+Since ˜c(s) and c(s) are related via the parameter s, one may view ˜c as a related vector field
+defined over ιs′,s′′(c(s)) ∈ Rn+1. Now the estimate in Theorem 1.1 simply asserts that the
+related vector fields ˜c(s) has an expansion module function ψ(t) = t with respect to the
+associated vector ιs′,s′′(c(s)).
+From the above connection between the concept of curvature and the expansion module
+it is our hope that a high dimensional Schur’s theorem could be discovered through the
+consideration involving the expansion module.
+Given that Schur’s theorem implies the Cauchy’s Arm Lemma for the arms of great arcs
+in the unit sphere, a natural question is that if the spherical analogue of Schur’s theorem
+still holds. Namely, given two embedded spherical curves c(s) and ˜c(s) in the unit sphere
+S2 ⊂ R3 parametrized by the arc-length s ∈ [0, L] with L ≤ π. Assume that c(s) is convex
+
+AN EXTENSION OF SCHUR’S THEOREM
+3
+with geodesic curvature k(s) > 0 and that the geodesic curvature of ˜c satisfies |˜k|(s) ≤ k(s).
+Does it still hold that |c(0) − c(L)| ≤ |˜c(0) − ˜c(L)|? One could also allow the tangent of
+curves to have same amount of finite many jumps at {sj}. In that case, the oriented angles
+αj and ˜αj are assumed to satisfy that αj ≥ ˜αj as in the case of Schur’s theorem. The
+Cauchy’s Arm Lemma in the sphere answers the question affirmatively in the special case
+where both curves have zero geodesic curvature for the smooth parts. Here we confirm this
+conjecture by proving
+Theorem 1.3. The Schur’s theorem holds for two curves in S2 ⊂ R3 under the above
+configurations similar to that of Theorem 1.1.
+The proof is via construction of auxiliary curves with one of them being a convex plane
+curve and appealing to the original Schur’s theorem. This is part of the reason we present
+the proof of Theorem 1.1 with care and details. Note that this result generalizes the spherical
+Cauchy’s Arm Lemma. It would be interesting to see if it plays any role in the proof of
+Pogorelov’s monotype theorem. There were extensions of A. Schur’s theorem in hyperbolic
+spaces [6] and in the Minkowski plane [8] earlier. It is plausible that the method of this
+paper can be used to simplify the argument in the former work via Theorem 2.1.
+2. Proof of Theorem 1.1
+We prove theorem and its corollary together. After an inclusion ι : R2 → Rn+1, which
+shall be specified later, we may consider the tangent T (s) and ˜T(s) as two curves in Sn. For
+the proof we need to choose a point N ∈ image(T (s)). For the situation when the tangent
+T (s) has a jump at sj, the minimizing arc jointing T (sj−) and T (sj+) is also considered
+to be part of the image. They together form a part of a great circle which is denoted by
+Image(T (s)).
+Consider the two curves T (s) and ˜T (s) inside Sn(1). The first one is a plane curve, hence
+is part of a great arc. We first find a s∗ ∈ [s′, s′′] (and then let N := T (s∗)), such that T (s∗)
+is parallel to c(s′′) − c(s′) using the convexity of the cone over image(T (s)).
+Since T (s), s ∈ [s′, s′′] forms a part of a great circle, letting j1 be the smallest j with sj ≥ s′
+and j2 being the greatest j with sj ≤ s′′, by the mean value theorem
+c(s′′) − c(s′)
+=
+� sj1
+s′
+T (s) ds +
+�
+j1≤j≤j2−1
+� sj+1
+sj
+T (s) ds +
+� s′′
+sj2
+T (s) ds
+=
+(sj1 − s′)T ((s∗)j1) +
+j2−1
+�
+j1
+(sj+1 − sj)T ((s∗)j+1) + (s′′ − sj2)T ((s∗)j2+1).
+Since
+1
+s′′−s′ multiple of the right hand above lies inside the cone over the image of T (s) for s ∈
+[s′, s′′], it implies that there exists (a unique) s∗ ∈ [s′, s′′] such that T (s∗) = λ(c(s′′) − c(s′))
+with λ =
+1
+|c(s′′)−c(s′)|.
+Now consider the two products Pi defined as (with N = T (s∗))
+P1 := ⟨c(s′′) − c(s′), N⟩ =
+� s′′
+s′ ⟨T (s), N⟩ ds,
+P2 := ⟨˜c(s′′) − ˜c(s′), N⟩ =
+� s′′
+s′ ⟨ ˜T(s), N⟩ ds.
+From the choice of s∗, ⟨c(s′′) − c(s′), N⟩ = |c(s′′) − c(s′)|. Now P1 = |c(s′′) − c(s′)|, and
+P2 = ⟨˜c(s′′) − ˜c(s′), c(s′′) − c(s′)⟩/|c(s′′) − c(s′)|.
+
+4
+LEI NI
+The claimed estimate (1.4) amounts to showing that the second product is bounded from
+below by the first after a proper inclusion. Let j3 be the biggest j with sj ≤ s∗. Observe
+the convexity of c(s) implies that
+αj0 +
+�
+j1≤j≤j3
+αj +
+� sj1
+s′
+k(s) ds +
+�
+j1≤j≤j3−1
+� sj+1
+sj
+k(s) ds +
+� s∗
+sj3
+k(s) ds = π,
+�
+j3+1≤j≤j2
+αj + αj4 +
+� sj3+1
+s∗
+k(s) ds +
+�
+j3+1≤j≤j2−1
+� sj+1
+sj
+k(s) ds +
+� s′′
+sj2
+k(s) ds = π,
+with αj0 being the angle from −N to T (s′) and αj4 being the angle from T (s′′) and −N.
+This implies that the images of T ([s′, s∗]) (denoted as curve Γ1(s)) and T ([s∗, s′′]) (denoted
+as the spherical curve Γ2(s)) are two minimizing arcs of the great circle (formed by the
+intersection of the plane in which c(s) lies and Sn(1)). We also denote the spherical curves
+corresponding to ˜T by �Γi. Hence
+max{Length(Γ1), Length(Γ2)} ≤ π.
+(2.1)
+On the other hand, by rotation we may arrange the inclusion ι such that ˜T(s∗) = N = T (s∗).
+Now we estimate
+π
+≥
+dSn(T (s′), N) = dSn(T (s′), T (s∗)) = Length(Γ1)
+=
+�
+j1≤j≤j3
+αj +
+� sj1
+s′
+k(s) ds +
+�
+j1≤j≤j3−1
+� sj+1
+sj
+k(s) ds +
+� s∗
+sj3
+k(s) ds
+≥
+�
+j1≤j≤j3
+˜αj +
+� sj1
+s′
+˜k(s) ds +
+�
+j1≤j≤j3−1
+� sj+1
+sj
+˜k(s) ds +
+� s∗
+sj3
+˜k(s) ds
+=
+�
+j1≤j≤j3
+˜αj +
+� sj1
+s′
+| ˜T ′|(s) ds +
+�
+j1≤j≤j3−1
+� sj+1
+sj
+| ˜T ′|(s) ds +
+� s′′
+sj3
+| ˜T ′|(s) ds
+≥
+dSn( ˜T(s′), ˜T(s∗)).
+The second line above follows from the definition of the curvature measuring the rotating
+angle of the tangent for a curve [10] (cf. page 49) and that for a convex curve k(s) ≥ 0. The
+third line uses the assumption, and the last line follows from the definition of the (spherical)
+distance between two points being the infimum of the length of all possible connecting curves
+in Sn. The same argument also implies that for any s ∈ [s′, s∗]
+π ≥ dSn(T (s), N) = Length(Γ1|[s,s∗]) ≥ dSn( ˜T(s), N).
+This implies that
+⟨T (s), N⟩ = cos(dSn(T (s), T (s∗)) ≤ cos(dSn( ˜T(s), ˜T (s∗)) = cos(dSn( ˜T(s), N)).
+(2.2)
+Rewriting the above estimate we have that
+⟨ ˜T (s) − T (s), N⟩ ≥ 0
+for s ∈ [s′, s∗], which implies (1.2). A similar argument proves that the same inequality
+holds also for s ∈ [s∗, s′′]. Putting them together we have (1.2). The above proof also works
+for any N = T (s∗) such that (2.1) holds, while (1.1) implies that one can always choose a
+s∗ ∈ [0, L] independent of s′ and s′′. (However such s∗ is far from being unique.)
+
+AN EXTENSION OF SCHUR’S THEOREM
+5
+Now we compare the two products Pi by writing
+P1 =
+� s′′
+s′ ⟨T (s), N⟩ ds =
+�� s∗
+s′
++
+� s′′
+s∗
+�
+cos (dSn(T (s), T (s∗)) ds.
+We express P2 accordingly. The above estimate (2.2) implies that
+� s∗
+s′
+cos(dSn(T (s), T (s∗)) ds ≤
+� s∗
+s′
+cos(dSn( ˜T (s), ˜T(s∗)) ds.
+(2.3)
+Similarly, we have
+� s′′
+s∗
+cos(dSn(T (s), T (s∗)) ds ≤
+� s′′
+s∗
+cos(dSn( ˜T (s), ˜T(s∗)) ds.
+(2.4)
+From (2.3) and (2.4) we have that P1 ≤ P2, namely the desired claim (1.4).
+From the proof we have the following more general monotonicity.
+Proposition 2.1. Let c : [0, L] → R2 be an embedded convex plane curve with curvature
+k(s) ≥ 0. Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s). Then there exists
+s∗ ∈ [0, L] and an inclusion ι : R2 → Rn+1 with ι(0) = 0 and ι(T (s∗)) = ˜T(s∗), such that
+I′
+1(s) = ⟨ ˜T (s) − ι(T (s)), ˜T(s∗)⟩ ≥ 0,
+∀ s ∈ [0, L], where I1(s) = ⟨˜c(s) − ι(c(s)), ˜T (s∗)⟩.
+(2.5)
+Here the choices of s∗ and ι are more flexible than in Theorem 1.1, where they are essentially
+unique.
+The proof can be easily adopted to show a comparison between a time-like curves in a
+Minkowski plane L2
+1 and another time-like curve in the three dimensional Minkowski space
+L3
+1 with signature (+, −, −). In fact in terms of the monotonicity one may choose s∗ freely.
+Following the convention of the physics a vector u is called time-like if ⟨u, u⟩ > 0. For a
+time-like curve c(s), |T (s)|2 = |c′(s)|2 = 1. Hence T (s) can be viewed as a point in the
+hyperbolic line/plane defined as x2
+1 − x2
+2 = 1 (or x2
+1 − x2
+2 − x2
+3 = 1). It can be checked easily
+that −1 multiple of the restricted metric on the surface is the standard hyperbolic metric.
+The angle between two tangents T (s1) and T (s2) is given by ⟨T (s1), T (s2)⟩ = cosh ϕ(s2, s1).
+A simple computation shows that ϕ(s2, s1) equals to the hyperbolic distance between T (s1)
+and T (s2).
+For space-like curves the length of the vector u is defined to be
+�
+−⟨u, u⟩.
+Equipped with the above basics, a similar consideration as the above gives the following
+result.
+Theorem 2.1. Let c(s) : [0, L] be a time-like convex curve in L2
+1 parametrized by the arc-
+length, and let ˜c(s) : [0, L] be a similarly parametrized regular time-like curve in L3
+1. Assume
+that k(s) ≥ |˜k|(s). Then for any s∗ ∈ [0, L] and an isometric inclusion of ι : L2
+1 → L3
+1, which
+identifies T (s∗) with ˜T(s∗), we have that
+I′
+2(s) = ⟨ι(T (s)) − ˜T(s), ˜T(s∗)⟩ ≥ 0, where I2(s) = ⟨ι(c(s)) − ˜c(s), ˜T (s∗)⟩.
+(2.6)
+In particular, |c(L) − c(0)| ≥ |˜c(L) − ˜c(0)|.
+The last statement of (ii) generalizes the result of [8] by allowing the second curve ˜c(s) a
+space curve in L3
+1. Note that for the curves in two Minkowski planes, the result for space-like
+curves is the same as that for the time-like curves. To prove the last conclusion we first
+integrate (2.6) with s∗ so chosen that T (s∗) is proportional to c(L) − c(0), and then apply
+the reserved Cauchy-Schwarz inequality (which holds for two time-like vectors).
+
+6
+LEI NI
+3. Proof of Theorem 1.3
+We start with some basics on spherical (smooth) curves.
+Let c(s) be a curve in S2
+parametrized by the arc-length. Let T (s) be its tangent, which is orthogonal to c(s). Let
+V (s) = c(s) × T (s) be the cross product of c(s) and T (s) in R3, which is a normal of c(s)
+in Tc(s)S2. The triple {c(s), T (s), V (s)} forms an orthonormal moving frame (of R3) along
+c(s). Since the geodesic curvature of a curve in the sphere (in a surface) is the changing
+rate of the tangential great circles (tangential geodesics in general, by (8-3) of page 157 of
+[13]), and that V (s) provides a natural parametrization of the tangential great circles, the
+derivative of V (s) yields the geodesic curvature of c(s). This can also be formulated in terms
+of the following result.
+Proposition 3.1. Let k(s) be the geodesic curvature of c(s) (with respect to S2). Then the
+following holds for {c(s), T (s), V (s)}.
+c′(s)
+=
+T (s),
+(3.1)
+T ′(s)
+=
+k(s)V (s) − c(s),
+(3.2)
+V ′(s)
+=
+−k(s)T (s).
+(3.3)
+Proof. The first equation is definition. Also by definition k(s) = ⟨T ′(s), V (s)⟩. Hence from
+0 = d2
+ds2
+�
+|c|2(s)
+�
+= 2⟨T (s), T (s)⟩ + 2⟨c(s), T ′(s)⟩ = 2 + 2⟨c(s), T ′(s)⟩
+we deduce the second equation. Now by the second equation
+V ′(s) = T (s) × T (s) + c(s) × T ′(s) = k(s) c(s) × V (s) = −k(s)T (s).
+This prove the third one, hence completes the proof of the proposition.
+□
+The local convexity of c(s) is equivalent to k(s) ≥ 0. The basic construction is to look
+at the cone C(c(s)) over the spherical curve c(s) centered at the origin, and obtain a plane
+curve by taking the intersection of C(c(s)) with a plane P not passing the origin to obtain
+a plane curve Pc(s). This curve can be expressed as R(s)c(s) with R(s) being the distance
+of Pc(s) to the origin. We need the following formula for the curvature of the space curve
+in R3 applied to Pc(s).
+Proposition 3.2. If c(s) is a convex curve in S2, Pc(s) is a convex curve in P.
+The
+curvature k(s) of Pc(s) (as a space curve of R3) is given by
+k2(s) = |P′
+c(s) × P′′
+c (s)|2
+|P′c(s)|3
+.
+(3.4)
+Proof. From the geometric definition of the convexity we know that c(s) lies in a signed
+semi-sphere cut out by any tangent great circle obtained by a plane passing the origin.
+Then it is clear that Pc(s) lies on the corresponding half plane cut out by the corresponding
+tangent line of Pc(s) in P. This proves the convexity of Pc(s). The formula for the curvature
+of a space curve is well known and computational. See for example page 51 of [10]. Of course
+the formula applies to the case that the curve happens to be a plane curve.
+□
+Now let τ be the arc-length parameter of Pc(s). Direct calculation shows that
+τ(s) =
+� s
+0
+�
+(R′(s))2 + R2(s) ds.
+(3.5)
+
+AN EXTENSION OF SCHUR’S THEOREM
+7
+Now we construct a space curve ˜P˜c(s) corresponding to ˜c(s) by defining it as R(s)˜c(s). In
+general, this is not a plane curve. The key observation is that the arc-length parameter for
+˜P˜c(s) is the same as that of Pc(s), namely it is given by (3.5) as well, since |˜c|(s) = 1 = |c(s)|
+and |˜c′|(s) = 1 = |c′(s)|. Moreover its curvature ˜k(s) (as a curve in R3) can be expressed
+similarly as
+˜k2(s) = | ˜P′
+˜c(s) × ˜P′′
+˜c (s)|2
+| ˜P′
+˜c(s)|3
+.
+(3.6)
+Namely the second part of Proposition 3.2 applies to ˜P˜c(s) as well since it holds for any
+space curve in R3. The key step is the following comparison.
+Proposition 3.3. Under the assumption that the geodesic curvature k(s) of c(s) and the
+geodesic curvature ˜k(s) of ˜c(s) satisfy k(s) ≥ |˜k(s)| ≥ 0, the curvatures of Pc(s) and ˜P˜c(s)
+satisfies k(s) ≥ 0 and k(s) ≥ |˜k|(s).
+Proof. Since Pc(s) is convex, we have that k(s) ≥ 0, it suffices to show that k2(s) ≥ ˜k2(s).
+First we observe that |P′
+c(s)|2 = R2(s) + (R′(s))2 = | ˜P′
+˜c(s)|2.
+This reduces the desired
+estimate to
+|P′
+c(s) × P′′
+c (s)|2 ≥ | ˜P′
+˜c(s) × ˜P′′
+˜c (s)|2.
+(3.7)
+Using the fact that {c(s), T (s), V (s)} forms an oriented orthonormal moving frame, a direct
+calculation, using Proposition 3.1, shows that
+P′
+c(s) × P′′
+c (s)
+=
+(R′(s)c(s) + R(s)T (s)) × (R′′(s)c(s) + 2R′(s)T (s) + R(s)T ′(s))
+=
+(2(R′(s))2 − R(s)R′′(s))V (s) − R′(s)R(s)k(s) T (s)
+−R(s)R′′(s) V (s) + R2(s)k(s)c(s) + R2(s)V (s)
+=
+R2(s)k(s)c(s) − R′(s)R(s)k(s) T (s)
++(2(R′(s))2 − 2R(s)R′′(s) + R2(s))V (s).
+Hence we have that
+|P′
+c(s) × P′′
+c (s)|2
+=
+(R4(s) + (R′(s)R(s))2)k2(s)
++(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2.
+(3.8)
+A similar calculation shows that
+| ˜P′
+˜c(s) × ˜P′′
+˜c (s)|2
+=
+(R4(s) + (R′(s)R(s))2)˜k2(s)
++(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2.
+(3.9)
+From (3.8) and (3.9), the assumption k(s) ≥ |˜k|(s) implies (3.7), hence the desired estimate
+of the proposition.
+□
+Now Proposition 3.3 and (3.5) implies that Pc(τ) and ˜P˜c(τ) are two curves satisfying the
+assumption of Theorem 1.1. Hence we have that
+d(Pc(0), Pc(τ(L))) ≤ d( ˜P˜c(0), ˜P˜c(τ(L))).
+Theorem 1.3 for the smooth curves now follows from the hinge theorem of Euclidean geom-
+etry.
+
+8
+LEI NI
+For the general case when the tangents of c(s) and ˜c(s) have finite jumps at {sj}, if we
+denote the turning angles at Pc(sj) and ˜P˜c(sj) by θj and ˜θj, then
+cos θj =
+R′(sj−)R′(sj+) + R2(sj) cos αj
+�
+((R′(sj−))2 + R2(sj)) ((R′(sj+))2 + R2(sj))
+.
+By a similar formula for cos ˜θj we deduce that θj ≥ ˜θj if αj ≥ ˜αj. Hence Theorem 1.3
+follows from the general case of Theorem 1.1.
+Acknowledgments
+The author would like to thank Burkhard Wilking for helpful discussions, Paul Bryant,
+Jon Wolfson, H. Wu and Fangyang Zheng for their interests to the problem considered.
+References
+[1] A. D. Alexandrow, Konvexe Polyeder. German translation from Russian; Akademie-Verlag, Berlin, 1958.
+[2] B. Andrews and J. Clutterbuck, Proof of the fundamental gap conjecture. J. Amer. Math. Soc. 24
+(2011), no. 3, 899–916.
+[3] H. Busemann, Convex surfaces. Interscience Tracts in Pure and Applied Mathematics, no. 6. Interscience
+Publishers, Inc., New York; Interscience Publishers Ltd., London 1958 ix+196 pp.
+[4] A. Cauchy, Sur les polygones et les poly`edres. Second M´emoire, Oeuvres Compl`etes, IIe S´erie, vol. 1;
+Paris, 1905.
+[5] S. S. Chern,
+Curves and surfaces in Euclidean space. 1967 Studies in Global Geometry and Analysis
+pp. 16–56 Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J.
+[6] C. L. Epstein, The theorem of A Schur in hyperbolic spaces. Preprint 46 pages, 1985.
+[7] H. Hopf, Differential Geometry in the Large. Notes taken by Peter Lax and John Gray. With a preface
+by S. S. Chern. Second edition. With a preface by K. Voss. Lecture Notes in Mathematics, 1000.
+Springer-Verlag, Berlin, 1989. viii+184 pp.
+[8] R. L´opez, The theorem of Schur in the Minkowski plane. Jour. Geom. Phys. 61 (2011), 342–346.
+[9] L. Ni, Estimates on the modulus of expansion for vector fields solving nonlinear equations. J. Math.
+Pures Appl. (9) 99 (2013), no. 1, 1–16.
+[10] A. V. Pogorelov, Differential Geometry. P. Noodhoff N. V. 1960.
+[11] E. Steinitz and H. Rademacher, Vorlesungen ¨uber die Th´eorie der Polyeder. Springer-Verlag, Berlin,
+1934.
+[12] I. J. Schoenberg and S. C. Zaremba,
+On Cauchy’s lemma concerning convex polygons. Canadian J.
+Math. 19 (1967), 1062–1071.
+[13] D. J. Struik, Lectures on Classical Differential Geometry. 2nd Edition, Dover, 1988.
+Lei Ni. Department of Mathematics, University of California, San Diego, La Jolla, CA 92093,
+USA
+Email address: leni@ucsd.edu
+

09E2T4oBgHgl3EQfNAa-/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf,len=334
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='03732v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='DG]  10 Jan 2023  A SCHUR’S THEOREM VIA A MONOTONICITY AND THE  EXPANSION MODULE  LEI NI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' In this paper we present a monotonicity which extends a classical theorem  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Schur comparing the chord length of a convex plane curve with a space curve of  smaller curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' We also prove a Schur’s Theorem for spherical curves, which extends  the Cauchy’s Arm Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Introduction  For a convex curve c(s) : [0, L] → R2 and a smooth curve in ˜c(s) : [0, L] → R3 of the same  length (both parametrized by the arc-length), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Schur’s theorem [7] (Theorem A page 31,  see also [5]) asserts that if both curves are embedded, and the curvature of the space curve  ˜k(s) := | ˜T ′|(s), where ˜T(s) = ˜c′(s) is the tangent vector, is not greater than the curvature  k(s) of the convex curve, then d(˜c(0), ˜c(L)) ≥ d(c(0), c(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' From the proof of [7] it is easy  to see R3 can be replaced by Rn with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The theorem can be proven for curves whose tangents have finite discontinuous jumps,  and to the situation that the curvature of the smaller curve is a curve in Rn+1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  In terms of the generalization to curves with finite discontinuous points for the tangent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' it  assumes that there exists {sj}0≤j≤N such that 0 = s0 < s1 < · · · < sk < · · · < sN = L such  that both c(s) and ˜c(s) are regular embedded curves for s ∈ (sj−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' sj) for all 1 ≤ j ≤ N  satisfying k(s) ≥ ˜k(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' and for each 1 ≤ j ≤ N − 1 at the point c(sj) and ˜c(sj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' the oriented  turning angles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' which are measured by signed distance αj := dSn(c′(sj−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' c′(sj+)) > 0 and  ˜αj = dSn(˜c′(sj−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' ˜c′(sj+)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' satisfy that αj ≥ ˜αj for all 1 ≤ j ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The convexity of  c(s) and the simpleness assumption imply that  N  �  j=1  � sj  sj−1  k(s) ds +  N−1  �  j=1  αj ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1)  This extension, together with some ingenious applications of the hinge’s theorem, allows  one to prove the famous Cauchy’s Arm Lemma for geodesic arms in the unit sphere (consist-  ing of continuous broken great/geodesic arcs with finite jumps of the tangents) in Lemma  II on the pages 37–38 of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The Lemma became famous due to that it had an incom-  plete/false proof by Cauchy originally [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The corrected proof appeared in [1, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' This  spherical Cauchy’s Arm Lemma can also be proved by an induction argument [12], whose  idea in fact in part resembles the proof of the smooth case to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Note that this  lemma of Cauchy plays a crucial role in the rigidity of convex polyhedra in R3, which finally  was vastly generalized to convex surfaces (convex bodies enclosed) as the famous Pogorelov  monotypy theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' [3] Section 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  1  2  LEI NI  The Schur’s theorem also can be applied to prove the four-vertex theorem for convex plane  curves, besides implying a Theorem of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Schwartz which asserts: For any curve c of  length L with curvature k(s) ≤ 1/r, let C be the circle passing c(0) and c(L) of radius r,  then L is either not greater than the length of the lesser circular arc, or not less than the  length of the greater circular arc of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  High dimensional (intrinsic) analogues of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Schur’s  theorem include the Rauch’s comparison theorem and the Toponogov comparison theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The later however has the limit of requiring that the manifold with less curvature must be  a space form of constant sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  First we have the following slight more general version of Schur’s theorem in terms of a  monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let c : [0, L] → R2 be an embedded convex plane curve with curvature  k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Then for any  0 ≤ s′ < s′′ ≤ L there exist an isometric inclusion ιs′,s′′ : R2 → Rn+1 with ιs′,s′′(0) = 0  such that  I(s) := ⟨˜c(s) − ιs′,s′′(c(s)), ιs′,s′′(c(s′′) − c(s′))⟩  is monotone non-decreasing for s ∈ [s′, s′′], or equivalently  ⟨ ˜T(s) − ιs′,s′′(T (s)), ιs′,s′′(c(s′′) − c(s′))⟩ ≥ 0,  ∀ s ∈ [s′, s′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2)  As s′ → s′′, the inclusion ιs′,s′′ converges to an inclusion identifying T (s) with ˜T(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Under the same assumption as in the theorem, for any s′ ≤ s′  ∗ < s′′  ∗ ≤ s′′,  ⟨c(s′′  ∗) − c(s′  ∗), c(s′′) − c(s′)⟩ ≤ ⟨˜c(s′′  ∗) − ˜c(s′  ∗), ιs′,s′′(c(s′′) − c(s′))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3)  When s′ = s′  ∗ and s′′ = s′′  ∗ we have that  |c(s′′) − c(s′)|2 ≤ ⟨˜c(s′′) − ˜c(s′), ιs′,s′′(c(s′′) − c(s′))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4)  The estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4) implies Schur’s theorem by the Cauchy-Schwarz inequality applied to  the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4):  |c(s′′) − c(s′)| ≤ |˜c(s′′) − ˜c(s′)|,  ∀ 0 ≤ s′ < s′′ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  This extension allows one to rephrase the result in terms of the concept of the expansion  module [2, 9] of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' If X : Ω ⊂ Rn+1 → Rn+1 is a vector field defined on a convex  domain, then the expansion module is a function of one variable ψ(t) such that  ⟨X(y) − X(x), y − x  |y − x|⟩ ≥ 2ψ  �|x − y|  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Since ˜c(s) and c(s) are related via the parameter s, one may view ˜c as a related vector field  defined over ιs′,s′′(c(s)) ∈ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Now the estimate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1 simply asserts that the  related vector fields ˜c(s) has an expansion module function ψ(t) = t with respect to the  associated vector ιs′,s′′(c(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  From the above connection between the concept of curvature and the expansion module  it is our hope that a high dimensional Schur’s theorem could be discovered through the  consideration involving the expansion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Given that Schur’s theorem implies the Cauchy’s Arm Lemma for the arms of great arcs  in the unit sphere, a natural question is that if the spherical analogue of Schur’s theorem  still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Namely, given two embedded spherical curves c(s) and ˜c(s) in the unit sphere  S2 ⊂ R3 parametrized by the arc-length s ∈ [0, L] with L ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Assume that c(s) is convex  AN EXTENSION OF SCHUR’S THEOREM  3  with geodesic curvature k(s) > 0 and that the geodesic curvature of ˜c satisfies |˜k|(s) ≤ k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Does it still hold that |c(0) − c(L)| ≤ |˜c(0) − ˜c(L)|?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' One could also allow the tangent of  curves to have same amount of finite many jumps at {sj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' In that case, the oriented angles  αj and ˜αj are assumed to satisfy that αj ≥ ˜αj as in the case of Schur’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The  Cauchy’s Arm Lemma in the sphere answers the question affirmatively in the special case  where both curves have zero geodesic curvature for the smooth parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Here we confirm this  conjecture by proving  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The Schur’s theorem holds for two curves in S2 ⊂ R3 under the above  configurations similar to that of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The proof is via construction of auxiliary curves with one of them being a convex plane  curve and appealing to the original Schur’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' This is part of the reason we present  the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1 with care and details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Note that this result generalizes the spherical  Cauchy’s Arm Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' It would be interesting to see if it plays any role in the proof of  Pogorelov’s monotype theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' There were extensions of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Schur’s theorem in hyperbolic  spaces [6] and in the Minkowski plane [8] earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' It is plausible that the method of this  paper can be used to simplify the argument in the former work via Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1  We prove theorem and its corollary together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' After an inclusion ι : R2 → Rn+1, which  shall be specified later, we may consider the tangent T (s) and ˜T(s) as two curves in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' For  the proof we need to choose a point N ∈ image(T (s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' For the situation when the tangent  T (s) has a jump at sj, the minimizing arc jointing T (sj−) and T (sj+) is also considered  to be part of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' They together form a part of a great circle which is denoted by  Image(T (s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Consider the two curves T (s) and ˜T (s) inside Sn(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The first one is a plane curve, hence  is part of a great arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' We first find a s∗ ∈ [s′, s′′] (and then let N := T (s∗)), such that T (s∗)  is parallel to c(s′′) − c(s′) using the convexity of the cone over image(T (s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Since T (s), s ∈ [s′, s′′] forms a part of a great circle, letting j1 be the smallest j with sj ≥ s′  and j2 being the greatest j with sj ≤ s′′, by the mean value theorem  c(s′′) − c(s′)  =  � sj1  s′  T (s) ds +  �  j1≤j≤j2−1  � sj+1  sj  T (s) ds +  � s′′  sj2  T (s) ds  =  (sj1 − s′)T ((s∗)j1) +  j2−1  �  j1  (sj+1 − sj)T ((s∗)j+1) + (s′′ − sj2)T ((s∗)j2+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Since  1  s′′−s′ multiple of the right hand above lies inside the cone over the image of T (s) for s ∈  [s′, s′′], it implies that there exists (a unique) s∗ ∈ [s′, s′′] such that T (s∗) = λ(c(s′′) − c(s′))  with λ =  1  |c(s′′)−c(s′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Now consider the two products Pi defined as (with N = T (s∗))  P1 := ⟨c(s′′) − c(s′), N⟩ =  � s′′  s′ ⟨T (s), N⟩ ds,  P2 := ⟨˜c(s′′) − ˜c(s′), N⟩ =  � s′′  s′ ⟨ ˜T(s), N⟩ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  From the choice of s∗, ⟨c(s′′) − c(s′), N⟩ = |c(s′′) − c(s′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Now P1 = |c(s′′) − c(s′)|, and  P2 = ⟨˜c(s′′) − ˜c(s′), c(s′′) − c(s′)⟩/|c(s′′) − c(s′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  4  LEI NI  The claimed estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4) amounts to showing that the second product is bounded from  below by the first after a proper inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let j3 be the biggest j with sj ≤ s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Observe  the convexity of c(s) implies that  αj0 +  �  j1≤j≤j3  αj +  � sj1  s′  k(s) ds +  �  j1≤j≤j3−1  � sj+1  sj  k(s) ds +  � s∗  sj3  k(s) ds = π,  �  j3+1≤j≤j2  αj + αj4 +  � sj3+1  s∗  k(s) ds +  �  j3+1≤j≤j2−1  � sj+1  sj  k(s) ds +  � s′′  sj2  k(s) ds = π,  with αj0 being the angle from −N to T (s′) and αj4 being the angle from T (s′′) and −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  This implies that the images of T ([s′, s∗]) (denoted as curve Γ1(s)) and T ([s∗, s′′]) (denoted  as the spherical curve Γ2(s)) are two minimizing arcs of the great circle (formed by the  intersection of the plane in which c(s) lies and Sn(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' We also denote the spherical curves  corresponding to ˜T by �Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Hence  max{Length(Γ1), Length(Γ2)} ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1)  On the other hand, by rotation we may arrange the inclusion ι such that ˜T(s∗) = N = T (s∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Now we estimate  π  ≥  dSn(T (s′), N) = dSn(T (s′), T (s∗)) = Length(Γ1)  =  �  j1≤j≤j3  αj +  � sj1  s′  k(s) ds +  �  j1≤j≤j3−1  � sj+1  sj  k(s) ds +  � s∗  sj3  k(s) ds  ≥  �  j1≤j≤j3  ˜αj +  � sj1  s′  ˜k(s) ds +  �  j1≤j≤j3−1  � sj+1  sj  ˜k(s) ds +  � s∗  sj3  ˜k(s) ds  =  �  j1≤j≤j3  ˜αj +  � sj1  s′  | ˜T ′|(s) ds +  �  j1≤j≤j3−1  � sj+1  sj  | ˜T ′|(s) ds +  � s′′  sj3  | ˜T ′|(s) ds  ≥  dSn( ˜T(s′), ˜T(s∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The second line above follows from the definition of the curvature measuring the rotating  angle of the tangent for a curve [10] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' page 49) and that for a convex curve k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The  third line uses the assumption, and the last line follows from the definition of the (spherical)  distance between two points being the infimum of the length of all possible connecting curves  in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The same argument also implies that for any s ∈ [s′, s∗]  π ≥ dSn(T (s), N) = Length(Γ1|[s,s∗]) ≥ dSn( ˜T(s), N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  This implies that  ⟨T (s), N⟩ = cos(dSn(T (s), T (s∗)) ≤ cos(dSn( ˜T(s), ˜T (s∗)) = cos(dSn( ˜T(s), N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2)  Rewriting the above estimate we have that  ⟨ ˜T (s) − T (s), N⟩ ≥ 0  for s ∈ [s′, s∗], which implies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' A similar argument proves that the same inequality  holds also for s ∈ [s∗, s′′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Putting them together we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The above proof also works  for any N = T (s∗) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1) holds, while (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1) implies that one can always choose a  s∗ ∈ [0, L] independent of s′ and s′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' (However such s∗ is far from being unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=')  AN EXTENSION OF SCHUR’S THEOREM  5  Now we compare the two products Pi by writing  P1 =  � s′′  s′ ⟨T (s), N⟩ ds =  �� s∗  s′  +  � s′′  s∗  �  cos (dSn(T (s), T (s∗)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  We express P2 accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The above estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2) implies that  � s∗  s′  cos(dSn(T (s), T (s∗)) ds ≤  � s∗  s′  cos(dSn( ˜T (s), ˜T(s∗)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3)  Similarly, we have  � s′′  s∗  cos(dSn(T (s), T (s∗)) ds ≤  � s′′  s∗  cos(dSn( ˜T (s), ˜T(s∗)) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4)  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4) we have that P1 ≤ P2, namely the desired claim (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  From the proof we have the following more general monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let c : [0, L] → R2 be an embedded convex plane curve with curvature  k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let ˜c : [0, L] → Rn+1 (n ≥ 1) be a curve such that ˜k(s) ≤ k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Then there exists  s∗ ∈ [0, L] and an inclusion ι : R2 → Rn+1 with ι(0) = 0 and ι(T (s∗)) = ˜T(s∗), such that  I′  1(s) = ⟨ ˜T (s) − ι(T (s)), ˜T(s∗)⟩ ≥ 0,  ∀ s ∈ [0, L], where I1(s) = ⟨˜c(s) − ι(c(s)), ˜T (s∗)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='5)  Here the choices of s∗ and ι are more flexible than in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1, where they are essentially  unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The proof can be easily adopted to show a comparison between a time-like curves in a  Minkowski plane L2  1 and another time-like curve in the three dimensional Minkowski space  L3  1 with signature (+, −, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' In fact in terms of the monotonicity one may choose s∗ freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Following the convention of the physics a vector u is called time-like if ⟨u, u⟩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' For a  time-like curve c(s), |T (s)|2 = |c′(s)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Hence T (s) can be viewed as a point in the  hyperbolic line/plane defined as x2  1 − x2  2 = 1 (or x2  1 − x2  2 − x2  3 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' It can be checked easily  that −1 multiple of the restricted metric on the surface is the standard hyperbolic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The angle between two tangents T (s1) and T (s2) is given by ⟨T (s1), T (s2)⟩ = cosh ϕ(s2, s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  A simple computation shows that ϕ(s2, s1) equals to the hyperbolic distance between T (s1)  and T (s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  For space-like curves the length of the vector u is defined to be  �  −⟨u, u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Equipped with the above basics, a similar consideration as the above gives the following  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let c(s) : [0, L] be a time-like convex curve in L2  1 parametrized by the arc-  length, and let ˜c(s) : [0, L] be a similarly parametrized regular time-like curve in L3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Assume  that k(s) ≥ |˜k|(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Then for any s∗ ∈ [0, L] and an isometric inclusion of ι : L2  1 → L3  1, which  identifies T (s∗) with ˜T(s∗), we have that  I′  2(s) = ⟨ι(T (s)) − ˜T(s), ˜T(s∗)⟩ ≥ 0, where I2(s) = ⟨ι(c(s)) − ˜c(s), ˜T (s∗)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='6)  In particular, |c(L) − c(0)| ≥ |˜c(L) − ˜c(0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The last statement of (ii) generalizes the result of [8] by allowing the second curve ˜c(s) a  space curve in L3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Note that for the curves in two Minkowski planes, the result for space-like  curves is the same as that for the time-like curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' To prove the last conclusion we first  integrate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='6) with s∗ so chosen that T (s∗) is proportional to c(L) − c(0), and then apply  the reserved Cauchy-Schwarz inequality (which holds for two time-like vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  6  LEI NI  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3  We start with some basics on spherical (smooth) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Let c(s) be a curve in S2  parametrized by the arc-length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let T (s) be its tangent, which is orthogonal to c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let  V (s) = c(s) × T (s) be the cross product of c(s) and T (s) in R3, which is a normal of c(s)  in Tc(s)S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The triple {c(s), T (s), V (s)} forms an orthonormal moving frame (of R3) along  c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Since the geodesic curvature of a curve in the sphere (in a surface) is the changing  rate of the tangential great circles (tangential geodesics in general, by (8-3) of page 157 of  [13]), and that V (s) provides a natural parametrization of the tangential great circles, the  derivative of V (s) yields the geodesic curvature of c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' This can also be formulated in terms  of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Let k(s) be the geodesic curvature of c(s) (with respect to S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Then the  following holds for {c(s), T (s), V (s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  c′(s)  =  T (s),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1)  T ′(s)  =  k(s)V (s) − c(s),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2)  V ′(s)  =  −k(s)T (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The first equation is definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Also by definition k(s) = ⟨T ′(s), V (s)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Hence from  0 = d2  ds2  �  |c|2(s)  �  = 2⟨T (s), T (s)⟩ + 2⟨c(s), T ′(s)⟩ = 2 + 2⟨c(s), T ′(s)⟩  we deduce the second equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Now by the second equation  V ′(s) = T (s) × T (s) + c(s) × T ′(s) = k(s) c(s) × V (s) = −k(s)T (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  This prove the third one, hence completes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  □  The local convexity of c(s) is equivalent to k(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The basic construction is to look  at the cone C(c(s)) over the spherical curve c(s) centered at the origin, and obtain a plane  curve by taking the intersection of C(c(s)) with a plane P not passing the origin to obtain  a plane curve Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' This curve can be expressed as R(s)c(s) with R(s) being the distance  of Pc(s) to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' We need the following formula for the curvature of the space curve  in R3 applied to Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' If c(s) is a convex curve in S2, Pc(s) is a convex curve in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  The  curvature k(s) of Pc(s) (as a space curve of R3) is given by  k2(s) = |P′  c(s) × P′′  c (s)|2  |P′c(s)|3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='4)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' From the geometric definition of the convexity we know that c(s) lies in a signed  semi-sphere cut out by any tangent great circle obtained by a plane passing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Then it is clear that Pc(s) lies on the corresponding half plane cut out by the corresponding  tangent line of Pc(s) in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' This proves the convexity of Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The formula for the curvature  of a space curve is well known and computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' See for example page 51 of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Of course  the formula applies to the case that the curve happens to be a plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  □  Now let τ be the arc-length parameter of Pc(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Direct calculation shows that  τ(s) =  � s  0  �  (R′(s))2 + R2(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='5)  AN EXTENSION OF SCHUR’S THEOREM  7  Now we construct a space curve ˜P˜c(s) corresponding to ˜c(s) by defining it as R(s)˜c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' In  general, this is not a plane curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The key observation is that the arc-length parameter for  ˜P˜c(s) is the same as that of Pc(s), namely it is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='5) as well, since |˜c|(s) = 1 = |c(s)|  and |˜c′|(s) = 1 = |c′(s)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Moreover its curvature ˜k(s) (as a curve in R3) can be expressed  similarly as  ˜k2(s) = | ˜P′  ˜c(s) × ˜P′′  ˜c (s)|2  | ˜P′  ˜c(s)|3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='6)  Namely the second part of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='2 applies to ˜P˜c(s) as well since it holds for any  space curve in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' The key step is the following comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Under the assumption that the geodesic curvature k(s) of c(s) and the  geodesic curvature ˜k(s) of ˜c(s) satisfy k(s) ≥ |˜k(s)| ≥ 0, the curvatures of Pc(s) and ˜P˜c(s)  satisfies k(s) ≥ 0 and k(s) ≥ |˜k|(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Since Pc(s) is convex, we have that k(s) ≥ 0, it suffices to show that k2(s) ≥ ˜k2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  First we observe that |P′  c(s)|2 = R2(s) + (R′(s))2 = | ˜P′  ˜c(s)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  This reduces the desired  estimate to  |P′  c(s) × P′′  c (s)|2 ≥ | ˜P′  ˜c(s) × ˜P′′  ˜c (s)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='7)  Using the fact that {c(s), T (s), V (s)} forms an oriented orthonormal moving frame, a direct  calculation, using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1, shows that  P′  c(s) × P′′  c (s)  =  (R′(s)c(s) + R(s)T (s)) × (R′′(s)c(s) + 2R′(s)T (s) + R(s)T ′(s))  =  (2(R′(s))2 − R(s)R′′(s))V (s) − R′(s)R(s)k(s) T (s)  −R(s)R′′(s) V (s) + R2(s)k(s)c(s) + R2(s)V (s)  =  R2(s)k(s)c(s) − R′(s)R(s)k(s) T (s)  +(2(R′(s))2 − 2R(s)R′′(s) + R2(s))V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Hence we have that  |P′  c(s) × P′′  c (s)|2  =  (R4(s) + (R′(s)R(s))2)k2(s)  +(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='8)  A similar calculation shows that  | ˜P′  ˜c(s) × ˜P′′  ˜c (s)|2  =  (R4(s) + (R′(s)R(s))2)˜k2(s)  +(2(R′(s))2 − 2R(s)R′′(s) + R2(s))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='9)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='9), the assumption k(s) ≥ |˜k|(s) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='7), hence the desired estimate  of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  □  Now Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='5) implies that Pc(τ) and ˜P˜c(τ) are two curves satisfying the  assumption of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Hence we have that  d(Pc(0), Pc(τ(L))) ≤ d( ˜P˜c(0), ˜P˜c(τ(L))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3 for the smooth curves now follows from the hinge theorem of Euclidean geom-  etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  8  LEI NI  For the general case when the tangents of c(s) and ˜c(s) have finite jumps at {sj}, if we  denote the turning angles at Pc(sj) and ˜P˜c(sj) by θj and ˜θj, then  cos θj =  R′(sj−)R′(sj+) + R2(sj) cos αj  �  ((R′(sj−))2 + R2(sj)) ((R′(sj+))2 + R2(sj))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  By a similar formula for cos ˜θj we deduce that θj ≥ ˜θj if αj ≥ ˜αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='3  follows from the general case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Acknowledgments  The author would like to thank Burkhard Wilking for helpful discussions, Paul Bryant,  Jon Wolfson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Wu and Fangyang Zheng for their interests to the problem considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content=' Interscience  Publishers, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=', New York;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Interscience Publishers Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=', London 1958 ix+196 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content=' Cauchy, Sur les polygones et les poly`edres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Second M´emoire, Oeuvres Compl`etes, IIe S´erie, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Paris, 1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content=' Chern,  Curves and surfaces in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' 1967 Studies in Global Geometry and Analysis  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' 16–56 Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' America, Buffalo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' distributed by Prentice-Hall, Englewood Cliffs, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Epstein, The theorem of A Schur in hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Preprint 46 pages, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Hopf, Differential Geometry in the Large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Notes taken by Peter Lax and John Gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' With a preface  by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Chern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' With a preface by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Voss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Lecture Notes in Mathematics, 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Springer-Verlag, Berlin, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' viii+184 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content=' Jour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content=' Canadian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+page_content='  [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Struik, Lectures on Classical Differential Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' 2nd Edition, Dover, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content='  Lei Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
+page_content=' Department of Mathematics, University of California, San Diego, La Jolla, CA 92093,  USA  Email address: leni@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E2T4oBgHgl3EQfNAa-/content/2301.03732v1.pdf'}
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+Imitator: Personalized Speech-driven 3D Facial Animation
+Balamurugan Thambiraja1
+Ikhsanul Habibie2
+Sadegh Aliakbarian3
+Darren Cosker3
+Christian Theobalt2
+Justus Thies1
+1 Max Planck Institute for Intelligent Systems, T¨ubingen, Germany
+2 Max Planck Institute for Informatics, Saarland, Germany
+3 Microsoft Mixed Reality & AI Lab, Cambridge, UK
+Figure 1. Imitator is a novel method for personalized speech-driven 3D facial animation. Given an audio sequence and a personalized
+style-embedding as input, we generate person-specific motion sequences with accurate lip closures for bilabial consonants (’m’,’b’,’p’).
+The style-embedding of a subject can be computed by a short reference video (e.g., 5s).
+Abstract
+Speech-driven 3D facial animation has been widely ex-
+plored, with applications in gaming, character animation,
+virtual reality, and telepresence systems. State-of-the-art
+methods deform the face topology of the target actor to sync
+the input audio without considering the identity-specific
+speaking style and facial idiosyncrasies of the target ac-
+tor, thus, resulting in unrealistic and inaccurate lip move-
+ments. To address this, we present Imitator, a speech-driven
+facial expression synthesis method, which learns identity-
+specific details from a short input video and produces novel
+facial expressions matching the identity-specific speaking
+style and facial idiosyncrasies of the target actor. Specif-
+ically, we train a style-agnostic transformer on a large fa-
+cial expression dataset which we use as a prior for audio-
+driven facial expressions. Based on this prior, we optimize
+for identity-specific speaking style based on a short refer-
+ence video. To train the prior, we introduce a novel loss
+function based on detected bilabial consonants to ensure
+plausible lip closures and consequently improve the realism
+of the generated expressions. Through detailed experiments
+and a user study, we show that our approach produces tem-
+porally coherent facial expressions from input audio while
+preserving the speaking style of the target actors. Please
+check out the project page for the supplemental video and
+more results.
+1. Introduction
+3D digital humans raised a lot of attention in the past
+few years as they aim to replicate the appearance and mo-
+tion of real humans for immersive applications, like telep-
+resence in AR or VR, character animation and creation for
+entertainment (movies and games), and virtual mirrors for
+e-commerce.
+Especially, with the introduction of neural
+rendering [27, 28], we see immense progress in the photo-
+1
+arXiv:2301.00023v1  [cs.CV]  30 Dec 2022
+
+Imitator
+Personalized Style-Embedding
+Audio
+Bilabial Consonants
+Personalized 3D Facial Animation
+Timerealistic synthesis of such digital doubles [11,20,38]. These
+avatars can be controlled via visual tracking to mirror the fa-
+cial expressions of a real human. However, we need to con-
+trol the facial avatars with text or audio inputs for a series
+of applications. For example, AI-driven digital assistants
+rely on motion synthesis instead of motion cloning. Even
+telepresence applications might need to work with audio in-
+puts only, when the face of the person is occluded or cannot
+be tracked, since a face capture device is not available. To
+this end, we analyze motion synthesis for facial animations
+from audio inputs; note that text-to-speech approaches can
+be used to generate such audio. Humans are generally sen-
+sitive towards faces, especially facial motions, as they are
+crucial for communication (e.g., micro-expressions). With-
+out full expressiveness and proper lip closures, the gener-
+ated animation will be perceived as unnatural and implausi-
+ble. Especially if the person is known, the facial animations
+must match the subject’s idiosyncrasies.
+Recent methods for speech-driven 3D facial anima-
+tion [5, 10, 16, 21] are data-driven.
+They are trained on
+high-quality motion capture data and leverage pretrained
+speech models [13,23] to extract an intermediate audio rep-
+resentation. We can classify these data-driven methods into
+two categories, generalized [5,10,21] and personalized an-
+imation generation methods [16]. In contrast to those ap-
+proaches, we aim at a personalized 3D facial animation syn-
+thesis that can adapt to a new user while only relying on in-
+put RGB videos captured with commodity cameras. Specif-
+ically, we propose a transformer-based auto-regressive mo-
+tion synthesis method that predicts a generalized motion
+representation. This intermediate representation is decoded
+by a motion decoder which is adaptable to new users. A
+speaker embedding is adjusted for a new user, and a new
+motion basis for the motion decoder is computed.
+Our
+method is trained on the VOCA dataset [5] and can be ap-
+plied to new subjects captured in a short monocular RGB
+video. As lip closures are of paramount importance for bi-
+labial consonants (’m’,’b’,’p’), we introduce a novel loss
+based on the detection of bilabials to ensure that the lips
+are closed properly. We take inspiration from the locomo-
+tion synthesis field [14,18], where similar losses are used to
+enforce foot contact with the ground and transfer it to our
+scenario of physically plausible lip motions.
+In a series of experiments and ablation studies, we
+demonstrate that our method is able to synthesize facial ex-
+pressions that match the target subject’s motions in terms of
+style and expressiveness. Our method outperforms state-of-
+the-art methods in our metrical evaluation and user study.
+Please refer to our supplemental video for a detailed qual-
+itative comparison. In a user study, we confirm that per-
+sonalized facial expressions are important for the perceived
+realism.
+The contributions of our work Imitator are as follows:
+• a novel auto-regressive motion synthesis architec-
+ture that allows for adaption to new users by disen-
+tangling generalized viseme generation and person-
+specific motion decoding,
+• and a lip contact loss formulation for improved lip clo-
+sures based on physiological cues of bilabial conso-
+nants (’m’,’b’,’p’).
+2. Related Work
+Our work focuses on speech-driven 3D facial animation
+related to talking head methods that create photo-realistic
+video sequences from audio inputs.
+Talking Head Videos: Several prior works on speech-
+driven generation focus on the synthesis of 2D talking head
+videos. Suwajanakorn et al. [25] train an LSTM network on
+19h video material of Obama to predict his person-specific
+2D lip landmarks from speech inputs, which is then used for
+image generation. Vougioukas et al. [33] propose a method
+to generate facial animation from a single RGB image
+by leveraging a temporal generative adversarial network.
+Chung et al. [4] introduce a real-time approach to gener-
+ate an RGB video of a talking face by directly mapping the
+audio input to the video output space. This method can re-
+dub a new target identity not seen during training. Instead of
+performing direct mapping, Zhou et al. [39] disentangles the
+speech information in terms of speaker identity and content,
+allowing speech-driven generation that can be applied to
+various types of realistic and hand-drawn head portraits. A
+series of work [24,29,36,37] uses an intermediate 3D Mor-
+phable Model (3DMM) [2,8] to guide the 2D neural render-
+ing of talking heads from audio. Wang et al. [34] extend this
+work also to model the head movements of the speaker. Lip-
+sync3d [17] proposes data-efficient learning of personalized
+talking heads focusing on pose and lighting normalization.
+Based on dynamic neural radiance fields [11], Ad-nerf [12]
+and DFA-NeRF [35] learn personalized talking head mod-
+els that can be rendered under novel views, while being
+controlled by audio inputs. In contrast to these methods,
+our work focuses on predicting 3D facial animations from
+speech that can be used to drive 3D digital avatars with-
+out requiring retraining of the entire model to capture the
+person-specific motion style.
+Speech-Driven 3D Facial Animation: Speech-driven 3d
+facial animation is a vivid field of research. Earlier meth-
+ods [6, 7, 9, 15, 32] focus on animating a predefined facial
+rig using procedural rules. HMM-based models generate
+visemes from input text or audio, and the facial anima-
+tions are generated using viseme-dependent co-articulation
+models [6, 7] or by blending facial templates [15]. With
+recent advances in machine learning, data-driven meth-
+ods [3, 5, 10, 16, 21, 26, 29] have demonstrated their capa-
+bility to learn viseme patterns from data. These methods
+2
+
+Figure 2. Our architecture takes audio as input which is encoded by a pre-trained Wav2Vec2.0 model [1]. This audio embedding ˆa1:T is
+interpreted by an auto-regressive viseme decoder which generates a generalized motion feature ˆv1:T . A style-adaptable motion decoder
+maps these motion features to person-specific facial expressions ˆy1:T in terms of vertex displacements on top of a template mesh.
+are based on pretrained speech models [1, 13, 23] to gen-
+erate an abstract and generalized representation of the in-
+put audio, which is then interpreted by a CNN or auto-
+regressive model to map to either a 3DMM space or directly
+to 3D meshes. Karras et al. [16] learn a 3D facial animation
+model from 3-5 minutes of high-quality actor specific 3D
+data. VOCA [5] is trained on 3D data of multiple subjects
+and can animate the corresponding set of identities from in-
+put audio by providing a one-hot encoding during inference
+that indicates the subject. MeshTalk [21] is a generalized
+method that learns a categorical representation for facial
+expressions and auto-regressively samples from this cate-
+gorical space to animate a given 3D facial template mesh
+of a subject from audio inputs. FaceFormer [10] uses a
+pretrained Wav2Vec [1] audio representation and applies a
+transformer-based decoder to regress displacements on top
+of a template mesh. Like VOCA, FaceFormer provides a
+speaker identification code to the decoder, allowing one to
+choose from the training set talking styles. In contrast, we
+aim at a method that can adapt to new users, capturing their
+talking style and expressiveness.
+3. Method
+Our goal is to model person-specific speaking style and
+the facial idiosyncrasies of an actor, to generate 3D facial
+animations of the subject from novel audio inputs. As in-
+put, we assume a short video sequence of the subject which
+we leverage to compute the identity-specific speaking style.
+To enable fast adaptation to novel users without significant
+training sequences, we learn a generalized style-agnostic
+transformer on VOCAset [5]. This transformer provides
+generic motion features from audio inputs that are inter-
+pretable by a person-specific motion decoder.
+The mo-
+tion decoder is pre-trained and adaptable to new users via
+speaking style optimization and refinement of the motion
+basis. To further improve synthesis results, we introduce a
+novel lip contact loss based on physiological cues of the bi-
+labial consonants [7]. In the following, we will detail our
+model architecture and the training objectives and describe
+the style adaptation.
+3.1. Model Architecture
+Our architecture consists of three main components (see
+Figure 2): an audio encoder, a generalized auto-regressive
+viseme decoder, and an adaptable motion decoder.
+Audio Encoder: Following state-of-the-art motion synthe-
+sis models [5, 10], we use a generalized speech model to
+encode the audio inputs A. Specifically, we leverage the
+Wav2Vec 2.0 model [1]. The original Wav2Vec is based on
+a CNN architecture designed to produce a meaningful la-
+tent representation of human speech. To this end, the model
+is trained in a self-supervised and semi-supervised manner
+to predict the immediate future values of the current input
+speech by using a contrastive loss, allowing the model to
+learn from a large amount of unlabeled data. Wav2Vec 2.0
+extends this idea by quantizing the latent representation and
+incorporating a Transformer-based architecture [31]. We re-
+sample the Wav2Vec 2.0 output with a linear interpolation
+layer to match the sampling frequency of the motion (30fps
+for the VOCAset, with 16kHz audio), resulting in a con-
+textual representation {ˆa}T
+t=1 of the audio sequence for T
+motion frames.
+Auto-regressive Viseme Decoder: The decoder Fv takes
+the contextual representation of the audio sequence as input
+and produces style agnostic viseme features ˆvt in an auto-
+regressive manner. These viseme features describe how the
+lip should deform given the context audio and the previ-
+ous viseme features. In contrast to Faceformer [10], we
+propose to use of a classical transformer architecture [31]
+as viseme decoder, which learns the mapping from audio-
+3
+
+Start
+11
+Linear Deformation Basis
+Cross-Modal MH Attention
+Multi-Head Self-Attention
+ Speaker Identity
+Token
+Positional Encoding
+Feed Forward
+12
+Wav2Vec
+Linear
+Linear + Relu
+Linear + Relu
+Linear + Relu
+Style Linear
+Linear + Relu
+03
+Si
+y3
+DT
+A
+<V
+arT
+Audio Encoder
+Autoregressive Viseme Decoder
+Motion Decoderfeatures {ˆa}T
+t=1 to identity agnostic viseme features {ˆv}T
+t=1.
+The autoregressive viseme decoder is defined as:
+ˆvt = Fv(θv; ˆv1:t−1, ˆa1:T ),
+(1)
+where θv are the learnable parameters of the transformer.
+In contrast to the traditional neural machine translation
+(NMT) architectures that produce discrete text, our output
+representation is a continuous vector.
+NMT models use
+a start and end token to indicate the beginning and end
+of the sequence. During inference, the NMT model auto-
+regressively generates tokens until the end token is gener-
+ated. Similarly, we use a start token to indicate the begin-
+ning of the sequences. However, since the sequence length
+T is given by the length of the audio input, we do not use
+an end token. We inject temporal information into the se-
+quences by adding encoded time to the viseme feature in the
+sequence. We formulate the positionally encoded interme-
+diate representations ˆht as:
+ˆht = ˆvt + PE(t),
+(2)
+where PE(t) is a sinusoidal encoding function [31]. Given
+the sequence of positional encoded inputs ˆht, we use multi-
+head self-attention which generates the context representa-
+tion of the inputs by weighting the inputs based on their rel-
+evance. These context representations are used as input to a
+cross-modal multi-head attention block which also takes the
+audio features ˆa1:T from the audio encoder as input. A fi-
+nal feed-forward layer maps the output of this audio-motion
+attention layer to the viseme embedding ˆvt. In contrast to
+Faceformer [10], which feeds encoded face motions ˆyt to
+the transformer, we work with identity-agnostic viseme fea-
+tures which are independently decoded by the motion de-
+coder. We found that feeding face motions ˆyt via an in-
+put embedding layer to the transformer contains identity-
+specific information, which we try to avoid since we aim
+for a generalized viseme decoder that is disentangled from
+person-specific motion. In addition, using a general start
+token instead of the identity code [10] as the start token re-
+duces the identity bias further. Note that disentangling the
+identity-specific information from the viseme decoder im-
+proves the motion optimization in the style adaption stage
+of the pipeline (see Section 3.3), as gradients do not need to
+be propagated through the auto-regressive transformer.
+Motion Decoder: The motion decoder aims to generate 3D
+facial animation ˆy1:T from the style-agnostic viseme fea-
+tures ˆv1:T and a style embedding ˆSi. Specifically, our mo-
+tion decoder consists of two components, a style embedding
+layer and a motion synthesis block. For the training of the
+style-agnostic transformer and for pre-training the motion
+decoder, we assume to have a one-hot encoding of the iden-
+tities of the training set. The style embedding layer takes
+this identity information as input and produces the style
+embedding ˆSi, which encodes the identity-specific motion.
+The style embedding is concatenated with the viseme fea-
+tures ˆv1:T and fed into the motion synthesis block. The mo-
+tion synthesis block consists of non-linear layers which map
+the style-aware viseme features to the motion space defined
+by a linear deformation basis. During training, the deforma-
+tion basis is learned across all identities in the dataset. The
+deformation basis is fine-tuned for style adaptation to out-
+of-training identities (see Section 3.3). The final mesh out-
+puts ˆy1:T are computed by adding the estimated per-vertex
+deformation to the template mesh of the subject.
+3.2. Training
+Similar to Faceformer [10], we use an autoregressive
+training scheme instead of teacher-forcing to train our
+model on the VOCAset [5]. Given that VOCAset provides
+ground truth 3D facial animations, we define the following
+loss:
+Ltotal = λMSE · LMSE + λvel · Lvel + λlip · Llip,
+(3)
+where LMSE defines a reconstruction loss of the vertices,
+Lvel defines a velocity loss, and Llip measures lip contact.
+The weights are λMSE = 1.0, λvel = 10.0, and λlip = 5.0.
+Reconstruction Loss: The reconstruction loss LMSE is:
+LMSE =
+V
+�
+v=1
+Tv
+�
+t=1
+||yt,v − ˆyt,v||2,
+(4)
+where yt,v is the ground truth mesh at time t in sequence v
+(of V total sequences) and ˆyt,v is the prediction.
+Velocity Loss:
+Our motion decoder takes independent
+viseme features as input to produce facial expressions. To
+improve temporal consistency in the prediction, we intro-
+duce a velocity loss Lvel similar to [5]:
+Lvel =
+V
+�
+v=1
+Tv
+�
+t=2
+||(yt,v − yt−1,v) − (ˆyt,v − ˆyt−1,v)||2. (5)
+Lip Contact Loss: Training with LMSE guides the model
+to learn an averaged facial expression, thus resulting in im-
+proper lip closures. To this end, we introduce a novel lip
+contact loss for bilabial consonants (’m’,’b’,’p’) to improve
+lip closures.
+Specifically, we automatically annotate the
+VOCAset to extract the occurrences of these consonants;
+see Section 4. Using this data, we define the following lip
+loss:
+Llip =
+T
+�
+t=1
+N
+�
+j=1
+wt||yt,v − ˆyt,v||2,
+(6)
+where wt,v weights the prediction of frame t according to
+the annotation of the bilabial consonants. Specifically, wt,v
+is one for frames with such consonants and zero otherwise.
+4
+
+Note that for such consonant frames, the target yt,v repre-
+sents a face with a closed mouth; thus, this loss improves
+lip closures at ’m’,’b’ and ’p’s (see Section 5).
+3.3. Style Adaptation
+Given a video of a new subject, we reconstruct and track
+the face ˜y1:T (see Section 4). Based on this reference data,
+we first optimize for the speaker style-embedding ˆS and
+then jointly refine the linear deformation basis using the
+LMSE and Lvel loss. In our experiments, we found that this
+two-stage adaptation is essential for generalization to new
+audio inputs as it reuses the pretrained information of the
+motion decoder. As an initialization of the style embedding,
+we use a speaking style of the training set. We precompute
+all viseme features ˆv1:T once, and optimize the speaking
+style to reproduce the tracked faces ˜y1:T . We then refine
+the linear motion basis of the decoder to match the person-
+specific deformations (e.g., asymmetric lip motions).
+4. Dataset
+We train our method based on the VOCAset [5], which
+consists of 12 actors (6 female and 6 male) with 40 se-
+quences each with a length of 3 − 5 seconds. The dataset
+comes with a train/test set split which we use in our exper-
+iments. The test set contains 2 actors. The dataset offers
+audio and high-quality 3D face reconstructions per frame
+(60fps). For our experiment, we sample the 3D face recon-
+structions at 30fps. We train the auto-regressive transformer
+on this data using the loss from Equation (3). For the lip
+contact loss Llip, we automatically compute the labels as
+described below.
+To adapt the motion decoder to a new subject, we require
+a short video clip of the person. Using this sequence, we
+run a 3DMM-based face tracker to get the per-frame 3D
+shape of the person. Based on this data, we adapt the motion
+decoder as detailed in Section 3.3.
+Automatic Lip Closure Labeling: For the VOCAset, the
+transcript is available. Based on Wav2Vec features, we align
+the transcript with the audio track. As the lip closure is
+formed before we hear the bilabial consonants, we search
+for the lip closure in the tracked face geometry before the
+time-stamp of the occurrence of the consonants in the script.
+We show this process for a single sequence in Figure 3. The
+lip closure is detected by lip distance, i.e., the frame with
+minimal lip distance in a short time window before the con-
+sonant is assumed to be the lip closure.
+External Sequence Processing:
+We assume to have a
+monocular RGB video of about 2 minutes in length as input
+which we divide into train/validation/test sequences. Based
+on MICA [40], we estimate the 3D shape of the subject
+using the first frame of the video.
+Using this shape es-
+timate, we run an analysis-by-synthesis approach [30] to
+Figure 3. Automatic labeling of the bilabial consonants (’m’,’b’
+and ’p’) and their corresponding lip closures in a sequence of
+VOCAset [5]. We align the transcript with the audio track using
+Wav2vec [1] features and extract the time stamps for the bilabial
+consonants. To detect the lip closures for the bilabial consonants,
+we search for local-minima on the Lip distance curves (red). The
+lip loss weights wt,v in a window around the detected lip closure
+are set to fixed values of a Gaussian function. We show an example
+of detected lip closures in the figure (in the blue bounding box).
+estimate per-frame blendshape parameters of the FLAME
+3DMM [19]. Given these blendshape coefficients, we can
+compute the 3D vertices of the per-frame face meshes that
+we need to adapt the motion decoder. Note that in contrast
+to the training data of the transformer, we do not require
+any bilabial consonants labeling, as we adapt the motion
+decoder only based on the reconstruction and velocity loss.
+5. Results
+To validate our method, we conducted a series of qual-
+itative and quantitative evaluations, including a user study
+and ablation studies. For evaluation on the test set of VO-
+CAset [5], we randomly sample 4 sequences from the test
+subjects’ train set (each ∼ 5s long) and learn the speaking-
+style and facial idiosyncrasies of the subject via style adap-
+tation. We compare our method to the state-of-the-art meth-
+ods VOCA [5], Faceformer [10], and MeshTalk [21]. We
+use the original implementations of the authors. However,
+we found that MeshTalk cannot train on the comparably
+small VOCAset. Thus, we qualitatively compare against
+MeshTalk with their provided model trained on a large-scale
+proprietary dataset with 200 subjects and 40 sequences for
+each. Note that the pretrained MeshTalk model is not com-
+patible with the FLAME topology; thus, we cannot evaluate
+their method on novel identities. In addition to the experi-
+5
+
+Words Spoken: BAGPIPES AND BONGOS
+Time
+Audio
+GT Lip distance curve
+Lip loss Weight
+Local Minimum search
+x- Detected consonants
+ × - Lip closure computedFigure 4. Qualitative comparison to the state-of-the-art methods VOCA [5], Faceformer [10], and MeshTalk [21]. Note that MeshTalk is
+performed with a different identity since we use their pretrained model, which cannot be trained on VOCAset. As we see in the highlighted
+regions, the geometry of the generated sequences without the person-specific style have muted and inaccurate lip animations.
+ments on the VOCAset, we show results on external RGB
+sequences. The results can be best seen in the suppl. video.
+Quantitative Evaluation: To quantitatively evaluate our
+Method
+Lface
+2
+↓
+Llip
+2
+↓
+F-DTW ↓
+Lip-DTW ↓
+Lip-sync ↓
+VOCA [5]
+0.88
+0.15
+1.28
+2.41
+5.72
+Faceformer [10]
+0.8
+0.14
+1.18
+2.85
+5.41
+Ours (w/ 1seq)
+0.91
+0.1
+1.3
+1.68
+3.99
+Ours
+0.89
+0.09
+1.26
+1.47
+3.78
+Table 1. Quantitative results on the VOCAset [5]. Our method
+outperforms the baselines on all of the lip metrics while perform-
+ing on par on the full-face metrics. Note that we are not targeting
+the animation of the upper face but aim for expressive and accurate
+lip movements, which is noticeable from the improved lip scores.
+method, we use the test set of VOCAset [5], which provides
+high-quality reference mesh reconstructions. We evaluate
+the performance of our method based on a mean L2 ver-
+tex distance for the entire mesh Lface
+2
+and the lip region
+Llip
+2 . Following MeshTalk [21], we also compute the Lip-
+sync, which measures the mean of the maximal per-frame
+lip distances. In addition, we use Dynamic Time Wrapping
+(DTW) to compute the similarity between the produced and
+reference meshes, both for the entire mesh (F-DTW) and the
+lip region (Lip-DTW). Since VOCA and Faceformer do not
+adapt to new user talking styles, we select the talking style
+from their training with the best quantitative metrics. Note
+that the pretrained MeshTalk model is not applicable to this
+6
+
+Words spoken
+So, I start talking now.... usually..
+One of my favorite topics to discuss is .
+Time
+0.0
+1.0
+1.5
+2.0
+2.5
+0.0
+1.0
+1.5
+2.0
+2.5
+GT
+Tracked GT
+Ours
+Faceformer
+VOCA
+MeshtalkFigure 5. Qualitative ablation comparison. At first, we show that our complete method with style and Llip loss is able to generate
+personalized facial animation with expressive motion and accurate lip closures. Replacing the person-specific style with the style seen
+during training results in generic and muted facial animation. As highlighted in the per-vertex error maps (magenta), the generated
+expression is not similar to the target actor. Especially the facial deformations are missing person-specific details. Removing Llip from the
+training objective results in improper lip closures (red).
+7
+
+Words spoken
+His Failure to Open ... By Job.
+Had Vinyl Technology Expand..
+Time
+0.0
+1.0
+1.5
+2.0
+2.5
+0.0
+1.0
+1.5
+2.0
+2.5
+香香香香香香香香香香
+GT
+香香香香香香香香香香
+Ours w/
+Sty + Lip
+Per-vertex
+Error (mm)
+0.0
+10
+Ours w/
+Train Sty 01
++ Lip
+Ours w/
+Train Sty 02
++ Lip
+Ours w/
+Sty + No Lip
+I Lip Closure
+error
+LMethod
+Expressiveness (%)
+Realism/Lip-sync (%)
+Ours vs VOCA [5]
+86.48
+76.92
+Ours vs Faceformer [10]
+81.89
+75.46
+Ours vs Ground truth
+20.28
+42.30
+Table 2. In a perceptual A/B user study conducted on the test set
+of VOCAset [5] with 56 participants, we see that in comparison to
+VOCA [5] and Faceformer [10] our method is preferred.
+evaluation due to the identity mismatch. As can be seen
+in Table 1, our method achieves the lowest lip reconstruc-
+tion and lip-sync errors, confirming our qualitative results.
+Even when using a single reference video for style adapta-
+tion (5s), our results shows significantly better lip scores.
+Qualitative Evaluation: We conducted a qualitative eval-
+uation on external sequences not part of VOCAset. In Fig-
+ure 4, we show a series of frames from those sequences
+with the corresponding words. As we can see, our method
+is able to adapt to the speaking style of the respective
+subject.
+VOCA [5] and Faceformer [10] miss person-
+specific deformations and are not as expressive as our re-
+sults. MeshTalk [21], which uses an identity that comes
+with the pretrained model, also shows dampened expressiv-
+ity. In the suppl. video, we can observe that our method is
+generating better lip closures for bilabial consonants.
+Perceptual Evaluation: We conducted a perceptual evalu-
+ation to quantify the quality of our method’s generated re-
+sults (see Table 2). Specifically, we conducted an A/B user
+study on the test set of VOCAset. We randomly sample 10
+sequences of the test subjects and run our method, VOCA,
+and Faceformer. For VOCA and Faceformer, which do not
+adapt to the style of a new user, we use the talking style
+of the training Subject 137, which provided the best quan-
+titative results. We use 20 videos per method resulting in
+60 A/B comparisons. For every A/B test, we ask the user
+to choose the best method based on realism and expressive-
+ness, following the user study protocol of Faceformer [10].
+In Table 2, we show the result of this study in which 56
+people participated. We observe that our method consis-
+tently outperforms VOCA and Faceformer. We also see that
+our model achieves similar realism and lip-sync as ground
+truth. Note that the users in the perceptual study have not
+seen the original talking style of the actors before. How-
+ever, the results show that our personalized synthesis leads
+to more realistic-looking animations.
+5.1. Ablation Studies
+To understand the impact of our style adaptation and
+the novel lip contact loss Llip on the perceptual quality,
+we show a qualitative ablation study including per-vertex
+error maps in Figure 5. As highlighted in the figure, the
+style adaptation is critical to match the person-specific de-
+formations and mouth shapes and improves expressiveness.
+Figure 6. Analysis of style adaptation in terms of lip distance on
+a test sequence of the VOCAset [5] (reference in red). Starting
+from an initial talking style from the training set (blue), we con-
+secutively adapt the style code (green) and the motion basis of the
+motion decoder (purple).
+The lip contact loss improves the lip closures for the bi-
+labial consonants, thus, improving the perceived realism,
+as can best be seen in the suppl. video. We rely on only
+∼ 60 seconds-long reference videos to extract the person-
+specific speaking style. A detailed analysis of the sequence
+length’s influence on the final output quality can be found
+in the suppl. material. It is also worth noting that our style-
+agnostic architecture allows us to perform style adaptation
+of the motion decoder in less than 30min, while an adapta-
+tion with an identity-dependent transformer takes about 6h.
+Our proposed style adaptation has two stages as ex-
+plained in Section 3.3. In the first step, we optimize for
+the style code and the refine the motion basis. In Figure 6,
+we show an example of the style adaptation by evaluating
+the lip distances throughout a sequence with a motion de-
+coder at initialization, with optimized style code, and with a
+refined motion basis. While the lip distance with the gener-
+alized motion decoder is considerable, it gets significantly
+improved by the consecutive steps of style adaptation. Af-
+ter style code optimization, we observe that the amplitude
+and frequency of the lip distance curves start resembling the
+ground truth. Refining the motion basis further improves
+the lip distance, and it is able to capture facial idiosyn-
+crasies, like asymmetrical lip deformations.
+6. Discussion
+Our evaluation shows that our proposed method outper-
+forms state-of-the-art methods in perceived expressiveness
+and realism. However, several limitations remain. Specifi-
+cally, we only support the speaking style of the subject seen
+in the reference video and do not control the talking style
+w.r.t. emotions (e.g., sad, happy, angry). The viseme trans-
+former and the motion decoder could be conditioned on an
+emotion flag; we leave this for future work. The expressive-
+ness and facial details depend on the face tracker’s quality;
+if the face tracking is improved, our method will predict
+better face shapes.
+8
+
+Speaking-Style Adaption
+Initial Style
+Lip distance (in mm)
+16
+Style code optimization
+Motion basis refinement
+14
+GT
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+110
+Time (in Frame steps)7. Conclusion
+We present Imitator, a novel approach for personalized
+speech-driven 3D facial animation. Based on a short refer-
+ence video clip of a subject, we learn a personalized motion
+decoder driven by a generalized auto-regressive transformer
+that maps audio to intermediate viseme features. Our stud-
+ies show that personalized facial animations are essential for
+the perceived realism of a generated sequence. Our new loss
+formulation for accurate lip closures of bilabial consonants
+further improves the results. We believe that personalized
+facial animations are a stepping stone towards audio-driven
+digital doubles.
+8. Acknowledgements
+This project has received funding from the Mesh Labs,
+Microsoft, Cambridge, UK. Further, we would like to thank
+Berna Kabadayi, Jalees Nehvi, Malte Prinzler and Wojciech
+Zielonka for their support and valuable feedback. The au-
+thors thank the International Max Planck Research School
+for Intelligent Systems (IMPRS-IS) for supporting Balamu-
+rugan Thambiraja.
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+
+Imitator: Personalized Speech-driven 3D Facial Animation
+– Supplemental Document –
+9. Impact of Data to Style Adaptation:
+To analyze the impact of data on the style adaptation pro-
+cess, we randomly sample (1, 4, 10, 20) sequences from the
+train set of the VOCA test subjects and perform our style
+adaption. Each sequence contains about 3 − 5 seconds of
+data. In Table 3, we observe that the performance on the
+quantitative metrics increase with the number of reference
+sequences. As mentioned in the main paper, even an adapta-
+tion based on a single sequence results in a significantly bet-
+ter animation in comparison to the baseline methods. This
+highlights the impact of style on the generated animations.
+Figure 7 illustrates the lip distance curve for one test se-
+quence used in this study. We observe that the lip distance
+with more reference data better fits the ground truth curve.
+No. Seq.
+Lface
+2
+↓
+Llip
+2
+↓
+F-DTW ↓
+Lip-DTW ↓
+Lip-sync ↓
+1
+0.91
+0.1
+1.3
+1.68
+3.99
+4
+0.89
+0.1
+1.26
+1.47
+3.78
+10
+0.76
+0.09
+1.07
+1.37
+3.57
+20
+0.7
+0.09
+0.99
+1.27
+3.49
+Table 3. Ablation of the style adaptation w.r.t. the amount of ref-
+erence sequences used. With an increasing number of data, the
+quantitative metrics improve. Each sequence is 3 − 5s long.
+Figure 7. With an increasing number of reference data samples
+for style adaptation, the lip distance throughout a test sequence of
+VOCAset is approaching the ground truth lip distance curve.
+10. Architecture Details
+10.1. Audio Encoder:
+Similar to Faceformer [10], our audio encoder is built
+upon the Wav2Vec 2.0 [1] architecture to extract temporal
+audio features. These audio features are fed into a linear in-
+terpolation layer to convert the audio frequency to the mo-
+tion frequency. The interpolated outputs are then fed into 12
+identical transformer encoder layers with 12 attention heads
+and an output dimension of 768. A final linear projection
+layer converts the audio features from the 768-dimension
+features to a 64-dimensional phoneme representation.
+10.2. Auto-regressive Viseme Decoder:
+Our auto-regressive viseme decoder is built on top of tra-
+ditional transformer decoder layers [31]. We use a zero
+vector of 64-dimension as a start token to indicate the start
+of sequence synthesis. We first add a positional encoding
+of 64-dimension to the input feature and fed it to decoder
+layers in the viseme decoder. For self-attention and cross-
+modal multi-head attention, we use 4 heads of dimension
+64. Our feed forward layer dimension is 128.
+Multi-Head Self-Attention: Given a sequence of posi-
+tional encoded inputs ˆht, we use multi-head self-attention
+(self-MHA), which generates the context representation of
+the inputs by weighting the inputs based on their relevance.
+The Scaled Dot-Product attention function can be defined as
+mapping a query and a set of key-value pairs to an output,
+where queries, keys, values and outputs are vectors [31].
+The output is the weighted sum of the values; the weight is
+computed by a compatibility function of a query with the
+corresponding key. The attention can be formulated as:
+Attention(Q, K, V ) = σ(QKT
+√dk
+)V,
+(7)
+where Q, K, V are the learned Queries, Keys and Values,
+σ(·) denotes the softmax activation function, and dk is the
+dimension of the keys.
+Instead of using a single atten-
+tion mechanism and generating one context representation,
+MHA uses multiple self-attention heads to jointly generate
+multiple context representations and attend to the informa-
+tion in the different context representations at different po-
+sitions. MHA is formulated as follows:
+MHA(Q, K, V ) = [head1, ...., headh] · W O,
+(8)
+with headi = Attention(QW Q
+i , KW K
+i , V W V
+i ), where
+W O, W Q
+i , W K
+i , W V
+i are weights related to each input vari-
+able.
+Audio-Motion Multi-Head Attention The Audio-Motion
+Multi-Head attention aims to map the context representa-
+tions from the audio encoder to the viseme representations
+by learning the alignment between the audio and style-
+agnostic viseme features. The decoder queries all the exist-
+ing viseme features with the encoded audio features, which
+12
+
+Ablation No. of Seguence used for Style-Adaption
+GT
+20.0
+Lip distance (in mm)
+1 seq
+4 seq
+17.5
+10 seq
+20 seq
+7.5
+5.0
+2.5
+10
+20
+40
+50
+60
+30
+70
+80
+Time (in Frame steps)carry both the positional information and the contextual in-
+formation, thus, resulting in audio context-injected viseme
+features. Similar to Faceformer [10], we add an alignment
+bias along the diagonal to the query-key attention score to
+add more weight to the current time audio features. The
+alignment bias BA(1 ≤ i ≤ t, 1 ≤ j ≤ KT) is:
+BA(i, j) =
+�
+0
+if (i = j),
+−∞
+otherwise.
+(9)
+The modified Audio-Motion Attention is represented as:
+Attention(Qv, Ka, V a, BA) = σ(Qv(Ka)T
+√dk
++ BA)V a,
+(10)
+where Qv are the learned queries from viseme features, Ka
+the keys and V a the values from the audio features, σ(·) is
+the softmax activation function, and dk is the dimension of
+the keys.
+10.3. Motion Decoder:
+The motion decoder aims to generate 3D facial anima-
+tions ˆy1:T from the style-agnostic viseme features ˆv1:T and
+a style embedding ˆSi. Specifically, our motion decoder con-
+sists of two components, a style embedding layer and a mo-
+tion synthesis block. The style linear layer takes a one-hot
+encoder of 8-dimension and produce a style-embedding of
+64-dimension. The input viseme features are concatenated
+with the style-embedding and fed into 4 successive linear
+layers which have a leaky-ReLU as activation. The output
+dimension of the 4-layer block is 64 dimensional. A final
+fully connected layer maps the 64-dimension input features
+to the 3D face deformation described as per-vertex displace-
+ments of size 15069. This layer is defining the motion de-
+formation basis of a subject and is adapted based on a ref-
+erence sequence.
+Training Details: We use the ADAM optimizer with a
+learning rate of 1e-4 for both the style-agnostic trans-
+former training and the style adaptation stage. During the
+style-agnostic transformer training, the parameters of the
+Wave2Vec 2.0 layers in the audio encoder are fixed. Our
+model is trained for 300 epochs, and the best model is cho-
+sen based on the validation reconstruction loss. During the
+style-adaptation stage, we first generate the viseme features
+and keep them fixed during the style adaptation stage. Then,
+we optimize for the style embedding for 300 epochs. Fi-
+nally, the style-embedding and final motion deformation ba-
+sis is refined for another 300 epochs.
+11. Broader Impact
+Our proposed method aims at the synthesis of realistic-
+looking 3D facial animations. Ultimately, these animations
+can be used to drive photo-realistic digital doubles of people
+in audio-driven immersive telepresence applications in AR
+or VR. However, this technology can also be misused for
+so-called DeepFakes. Given a voice cloning approach, our
+method could generate 3D facial animations that drive an
+image synthesis method. This can lead to identity theft, cy-
+ber mobbing, or other harmful criminal acts. We believe
+that conducting research openly and transparently could
+raise the awareness of the misuse of such technology. We
+will share our implementation to enable research on digi-
+tal multi-media forensics. Specifically, synthesis methods
+are needed to produce the training data for forgery detec-
+tion [22].
+All participants in the study have given written consent
+to the usage of their video material for this publication.
+13
+

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+Rethinking the Video Sampling and Reasoning Strategies for
+Temporal Sentence Grounding
+Jiahao Zhu1∗, Daizong Liu2†∗, Pan Zhou1†, Xing Di3, Yu Cheng4, Song Yang5,
+Wenzheng Xu6, Zichuan Xu7, Yao Wan8, Lichao Sun9, Zeyu Xiong1
+1Huazhong University of Science and Technology 3ProtagoLabs Inc 4Microsoft Research
+2Peking University 5Beijing Institute of Technology 6School of Sichuan University
+7Dalian University of Technology 9Lehigh University
+8School of Computer Sci. & Tech., Huazhong University of Science and Technology
+{jiahaozhu, panzhou, wanyao, zeyuxiong}@hust.edu.cn, dzliu@stu.pku.edu.cn
+xing.di@protagolabs.com, yu.cheng@microsoft.com, S.Yang@bit.edu.cn,
+wenzheng.xu@scu.edu.cn, z.xu@dlut.edu.cn, lis221@lehigh.edu
+Abstract
+Temporal sentence grounding (TSG) aims to
+identify the temporal boundary of a specific
+segment from an untrimmed video by a sen-
+tence query.
+All existing works first uti-
+lize a sparse sampling strategy to extract a
+fixed number of video frames and then con-
+duct multi-modal interactions with query sen-
+tence for reasoning. However, we argue that
+these methods have overlooked two indispens-
+able issues: 1) Boundary-bias: The annotated
+target segment generally refers to two spe-
+cific frames as corresponding start and end
+timestamps. The video downsampling process
+may lose these two frames and take the adja-
+cent irrelevant frames as new boundaries. 2)
+Reasoning-bias: Such incorrect new bound-
+ary frames also lead to the reasoning bias
+during frame-query interaction, reducing the
+generalization ability of model. To alleviate
+above limitations, in this paper, we propose a
+novel Siamese Sampling and Reasoning Net-
+work (SSRN) for TSG, which introduces a
+siamese sampling mechanism to generate ad-
+ditional contextual frames to enrich and re-
+fine the new boundaries. Specifically, a rea-
+soning strategy is developed to learn the inter-
+relationship among these frames and generate
+soft labels on boundaries for more accurate
+frame-query reasoning.
+Such mechanism is
+also able to supplement the absent consecutive
+visual semantics to the sampled sparse frames
+for fine-grained activity understanding. Exten-
+sive experiments demonstrate the effectiveness
+of SSRN on three challenging datasets.
+1
+Introduction
+Temporal sentence grounding (TSG) is an impor-
+tant yet challenging task in natural language pro-
+∗Equal contributions.
+†Corresponding author.
+(a) An example of the temporal sentence grounding (TSG).
+Sentence Query: The woman then adds ginger ale, and shakes the drink in a tumbler.
+Ground Truth
+|
+|
+63.74s
+80.57s
+(a) An example of the temporal sentence grounding (TSG).
+Sentence Query: The woman then adds ginger ale, and shakes the drink in a tumbler.
+Ground Truth
+|
+|
+63.74s
+80.57s
+Video
+Original segment
+...
+...
+(b) An illustration of the video feature sampling of existing TSG methods.
+Video
+Original segment
+...
+...
+(b) An illustration of the video feature sampling of existing TSG methods.
+(c) An overview of our siamese sampling strategy.
+(c) An overview of our siamese sampling strategy.
+New segment
+New segment
+refine
+refine
+Siamese Sampling
+Siamese Sampling
+Original segment
+Original segment
+Contextual Frames
+Contextual Frames
+enrich
+enrich
+Contextual Frames
+Contextual Frames enrich
+enrich
+Sparse Sampling
+Sparse Sampling
+......
+Video
+Video
+......
+......
+......
+New segment
+New segment
+Refined segment
+Refined segment
+Figure 1: (a) An example of temporal sentence ground-
+ing task. (b) All existing TSG methods generally utilize
+a downsampling process to evenly extract a fixed num-
+ber of frames from a long video. However, the new tar-
+get segment is obtained by rounding operation and may
+introduces boundary bias since some original bound-
+ary frames are lost. (c) We propose a siamese sam-
+pling strategy to extract additional adjacent frames to
+enrich and refine the information of the sampled frames
+for generating more accurate boundary of the new seg-
+ment.
+cessing, which has drawn increasing attention over
+the last few years due to its vast potential applica-
+tions in information retrieval (Dong et al., 2019;
+Yang et al., 2020) and human-computer interaction
+(Singha et al., 2018). It aims to ground the most rel-
+evant video segment according to a given sentence
+query. As shown in Figure 1 (a), video and query
+information need to be deeply incorporated to dis-
+tinguish the fine-grained details of adjacent frames
+for determining accurate boundary timestamps.
+Previous TSG methods (Gao et al., 2017; Chen
+et al., 2018; Zhang et al., 2019b; Yuan et al., 2019a;
+arXiv:2301.00514v1  [cs.CV]  2 Jan 2023
+
+Zhang et al., 2020b; Liu et al., 2018a; Zhang et al.,
+2019a; Liu et al., 2018b, 2021a) generally follow
+an encoding-then-interaction framework that first
+extracts both video and query features and then con-
+duct multi-modal interactions for reasoning. Since
+many videos are overlong while corresponding tar-
+get segments are short, these methods simply uti-
+lize a sparse sampling strategy shown in Figure1
+(b), which samples a fixed number of frames from
+each video to reconstruct a shorter video, and then
+learn frame-query relations for segment inferring.
+We argue that existing learning paradigm suffers
+from two obvious limitations: 1) Boundary-bias:
+Each video has a query-related segment, which
+refers to two specific frames as its start and end
+timestamps. Traditional sparse downsampling strat-
+egy extracts frames from videos with a fixed in-
+terval. A rounding operation is then applied to
+map the annotated segment to the sampled frames
+by keeping the same proportional length in both
+original and new videos. As a result, the ground-
+truth boundary frames may be filtered out and the
+query-irrelevant frames will be regarded as the ac-
+tual boundaries, generating wrong labels for latter
+training. 2) Reasoning-bias: The query-irrelevant
+boundary frames in the newly reconstructed seg-
+ment will also lead to incorrect frame-query interac-
+tion and reasoning in the training process, reducing
+the generalization ability of model.
+To alleviate these two issues, a straightforward
+idea is to filter out the sampled boundary frames in
+the new segment if they are query-irrelevant. How-
+ever, this will destroy the true segment length when
+we transfer the downsampled segment back to the
+original one during the inference process. Another
+straightforward idea is to directly keep the appropri-
+ate segment length (by float values) in the newly re-
+constructed video and then reason the query content
+in the new boundary to determine what percentage
+of this boundary is correct. However, the query-
+irrelevant boundaries lack sufficient query-related
+information for boundary reasoning. Based on the
+above considerations, we aim to extract additional
+frames adjacent to the sampled frames to enrich
+and refine their information for supplementing the
+consecutive visual semantics. In this way, the new
+boundary frames are well semantic-correlated to
+its original adjacent boundaries. Based on the re-
+fined boundary frames, we can keep and learn the
+appropriate segment length of the downsampled
+video for query reasoning. Moreover, other inner
+frames are also enriched by their neighbors, captur-
+ing more consecutive visual appearances for fully
+understanding the entire activity.
+Therefore, in this paper, we propose a novel
+Siamese Sampling and Reasoning Network (SSRN)
+for temporal sentence grounding task to generate
+additional contextual frames to enrich and refine
+the new boundaries.
+Specifically, we treat the
+sparse sampled video frames as anchor frames, and
+additionally extract several frames adjacent to each
+anchor frame as the siamese frames for semantic
+sharing and enriching. A siamese knowledge ag-
+gregation module is designed to explore internal
+relationships and aggregate contextual information
+among these frames. Then, a siamese reasoning
+module supplements the fine-grained contexts of
+siamese frames into the anchor frames for enrich-
+ing their semantics. In this way, the query-related
+information are added into the new boundaries thus
+we can utilize an appropriate float value to rep-
+resent the new segment length for query reason-
+ing, addressing both boundary- and reasoning-bias.
+Moreover, other sampled frames are also equipped
+with more consecutive visual semantics from their
+original neighbors, which further benefits more
+fine-grained learning process.
+Our contributions are summarized as follows:
+• We propose a novel SSRN model which can
+sparsely extract multiple relevant frames from
+original videos to enrich the anchor frames
+for more accurate boundary prediction. To
+the best of our knowledge, we are the first to
+propose and address both boundary-bias and
+reasoning-bias in TSG task.
+• We propose an effective siamese aggregation
+and reasoning method to correlate and inte-
+grate the contextual information of siamese
+frames to refine the anchor frames.
+• Extensive experiments are conducted on three
+challenging public benchmarks, including Ac-
+tivityNet Captions, TACoS and Charades-
+STA, demonstrating the effectiveness of our
+proposed SSRN method.
+2
+Related Work
+Temporal sentence grounding (TSG) is a new task
+introduced recently (Gao et al., 2017; Anne Hen-
+dricks et al., 2017), which aims to localize the
+most relevant video segment from a video with
+
+sentence descriptions. All existing methods fol-
+low an encoding-then-interaction framework that
+first extracts video/query features and then conduct
+multi-modal interactions for segment inferring.
+Based on the interacted multi-modal features,
+traditional methods follow a propose-and-rank
+paradigm to make predictions. Most of them (Ge
+et al., 2019; Qu et al., 2020; Xiao et al., 2021; Liu
+et al., 2021a,c, 2020a; Liu and Hu, 2022a,b; Liu
+et al., 2022c; Fang et al., 2022; Liu et al., 2022f)
+typically utilize a proposal-based grounding head
+that first generates multiple candidate segments as
+proposals, and then ranks them according to their
+similarity with the query semantic to select the best
+matching one. Some of them (Gao et al., 2017;
+Anne Hendricks et al., 2017) directly utilize multi-
+scale sliding windows to produce the proposals and
+subsequently integrate the query with segment rep-
+resentations via a matrix operation. To improve the
+quality of the proposals, latest works (Wang et al.,
+2020; Yuan et al., 2019a; Zhang et al., 2019b; Cao
+et al., 2021; Liu et al., 2021b, 2020b, 2022d,e,a) in-
+tegrate sentence information with each fine-grained
+video clip unit, and predict the scores of candidate
+segments by gradually merging the fusion feature
+sequence over time.
+Recently, some proposal-free works (Yuan et al.,
+2019b; Wang et al., 2019; Rodriguez et al., 2020;
+Chen et al., 2020; Mun et al., 2020; Zeng et al.,
+2020; Zhang et al., 2020a, 2021; Nan et al., 2021)
+directly predict the temporal locations of the tar-
+get segment without generating complex proposals.
+These works directly select the starting and end-
+ing frames by leveraging cross-modal interactions
+between video and query. Specifically, they either
+regress the start/end timestamps based on the en-
+tire video representation (Yuan et al., 2019b; Mun
+et al., 2020), or predict at each frame to determine
+whether this frame is a start or end boundary (Ro-
+driguez et al., 2020; Chen et al., 2020; Zeng et al.,
+2020; Zhang et al., 2020a, 2021).
+Although the above two types of methods have
+achieved great performances, their video sampling
+strategy in encoding part is unreasonable that can
+lead to both boundary and reasoning bias. Specifi-
+cally, the boundary bias is defined as the incorrect
+boundary of the new segment reconstructed by the
+video sparse sampling. The reasoning bias is de-
+fined as the incorrect correlation learning between
+the query-irrelevant frames and query. In this pa-
+per, we aim to reduce the above bias by proposing
+a new siamese sampling and reasoning strategy to
+enrich the sampled frames and further refine the
+reconstructed segment boundary.
+3
+The Proposed Method
+Given an untrimmed video and a sentence query,
+we represent the video as V with a frame number
+of T. Similarly, the query with N words is denoted
+as Q. Temporal sentence grounding (TSG) aims
+to localize a segment (τs, τe) starting at timestamp
+τs and ending at timestamp τe in video V, which
+corresponds to the same semantic as query Q.
+The overall architecture of the proposed Siamese
+Sampling and Reasoning Network (SSRN) method
+is illustrated in Figure 2. The SSRN framework
+contains four main components: (1) Siamese sam-
+pling and encoding: We sparsely downsample
+each long video into the anchor frames, and a
+new siamese sampling strategy additionally sam-
+ples their adjacent frames as siamese frames. A
+video/query encoder then extracts visual/query fea-
+tures from all sampled video frames and query sen-
+tence respectively. (2) Multi-modal interaction: Af-
+ter that, we interact the query features with the vi-
+sual features for cross-modal interaction. (3) Multi-
+modal reasoning: Next, to supplement the knowl-
+edge of siamese frames into the anchor frames, a
+siamese knowledge aggregation module is devel-
+oped to determine how much the information of
+siamese frames are needed to inject into the an-
+chor ones. Then, a reasoning module is utilized
+to enrich the anchor frames with the aggregated
+semantic knowledge. In this way, the contexts of
+both new boundaries and other sparse frames are
+enriched and can better represent the full and con-
+secutive visual semantics. (4) Grounding heads
+with soft labels: At last, we employ the ground-
+ing heads with soft label to predict more accurate
+boundaries via float value to keep the appropriate
+segment length. We illustrate the details of each
+component in the following subsections.
+3.1
+Siamese Sampling and Encoding
+Given the dense video input V, previous works gen-
+erally downsample each video into a new video
+of fixed length to address the problem of overlong
+video. Considering the existing boundary-bias, we
+propose a siamese sampling strategy to additionally
+extract contextual adjacent frames nearby each sam-
+pled frame to enrich its query-related information
+for better determining the accurate new boundary.
+
+Xi 
+Video Input
+Multimodal 
+Interaction
+Rounded Boundary
+Predictor
+Float Boundary
+Predictor
+Siamese
+Knowledge
+Aggregation
+Siamese
+Knowledge
+Reasoning
+Video Frames
+Anchor frames
+Query Input
+Query Encoder
+Video Encoder
+Multi-Modal 
+Reasoning 
+������������������������, ������������������������
+������������������������, ������������������������
+Timeline
+������������������������
+{������������������������,������������}�������������=1
+������������−1
+������������������������
+{������������������������,������������}������������=1
+������������−1
+Q
+“The woman then adds ginger ale, 
+and shakes the drink in a tumbler.”
+Samping
+...
+...
+Siamese frames
+...
+...
+...
+length of  T
+length of  M
+length of  M
+������������
+������������
+Siamese
+Knowledge 
+Enriched anchor feature �������������������������
+̂������������������������′
+̂������������������������′
+������������������������( ̂������������������������′) ������������������������( ̂������������������������′)
+Soft
+Label
+length of  K
+������������������������
+{������������������������,������������}������������=1
+������������−1
+������������
+Figure 2: Overview of our Siamese Sampling and Reasoning Network. Given a dense video, the anchor frames
+and siamese frames are first extracted by sparse sampling and siamese sampling, respectively. Then a video/query
+encoder and a multimodal interaction module are utilized to generate multimodal features. Next, a siamese knowl-
+edge generation module is proposed to model contextual relationship between anchor frames and siamese ones
+from the same video. After that, the siamese knowledge reasoning module exploits the siamese knowledge to en-
+rich the information of the anchor frames for more accurate boundary prediction. At last, in the grounding heads,
+we utilize a soft label to learn more fine-grained boundaries of float value in addition to the rounded one.
+Here, we call the downsampled frames and their
+contextual frames as anchor frames and siamese
+frames, respectively.
+Specifically, as shown in
+Figure 1 (c), following previous works, we di-
+rectly construct the anchor video Va by sparsely
+and evenly sampling M frames from dense video
+frames of length T (T is usually much greater
+than M). The new siamese videos are then cap-
+tured at different beginning indices in the original
+video but next to the frames of the anchor video.
+The same sample interval is utilized for all frames.
+After siamese sampling, we can obtain multiple
+siamese videos with same length and similar global
+semantics as the anchor video. We denote the new
+siamese videos as {Vs,k}K
+k=1 where K means the
+siamese sample number.
+Since we utilize the sampling strategy to pro-
+cess the dense video frames, the start/end time
+of the target segment in original video sequence
+needs to be accurately mapped to the correspond-
+ing boundaries in the new video sequence of M
+frames. Following almost all previous TSG meth-
+ods (Zhang et al., 2019b, 2020a; Liu et al., 2021a),
+the new start/end index is generally calculated by
+ˆτs(e) = ⌊τs(e)/T × M⌋, where ⌊·⌋ denotes the
+rounding operator. During the inference, the pre-
+dicted segment boundary index can be easily con-
+verted to the corresponding time in the dense video
+via τs(e) = ˆτs(e)/M × T. However, the rounding
+operation may produce boundary bias that the new
+boundary frames are not semantically correlated to
+the query semantic. Therefore, we further generate
+a soft label ˜τs(e) = ⟨τs(e)/T × M⟩ as an additional
+supervision to keep the appropriate segment length
+during training, where ⟨·⟩ denotes the float result.
+Video encoder
+For video encoding, we first ex-
+tract frame features by a pre-trained C3D network
+(Tran et al., 2015), and then add a positional en-
+coding (Vaswani et al., 2017) to provide positional
+knowledge. Such position encoding plays a crucial
+role in distinguishing semantics at diverse temporal
+locations. Considering the sequential characteristic
+in videos, a Bi-GRU (Chung et al., 2014) is further
+applied to incorporate the contextual information
+along time series. We denote the extracted video
+features of both anchor video and siamese video as
+V a, {V s,k}K
+k=1 ∈ RM×D, respectively.
+Query encoder For query encoding, we first ex-
+tract word embeddings by the Glove model (Pen-
+nington et al., 2014). We also apply positional
+encoding and Bi-GRU to integrate the sequential
+information within the sentence. The final feature
+of the query is denoted as Q ∈ RN×D.
+
+3.2
+Multi-Modal Interaction
+After obtaining the video features V a, {V s,k}K
+k=1
+and query feature Q, we utilize a co-attention mech-
+anism (Lu et al., 2019) to capture the cross-modal
+interactions between them. Specifically, for each
+video feature V ∈ {V a} ∪ {V s,k}K
+k=1, we first
+calculate the similarity between V and Q as:
+S = V (QWS)⊤ ∈ RM×N,
+(1)
+where WS ∈ RD×D projects the query features
+into the same latent space as the video. Then, we
+compute two attention weights as:
+A = Sr(QWS) ∈ RM×D,
+B = SrST
+c V ∈ RM×D,
+(2)
+where Sr and Sc are the row- and column-wise
+softmax results of S, respectively. We compose the
+final query-guided video representation by learning
+its sequential features as follows:
+F = Bi-GRU([V ; A; V ⊙A; V ⊙B]) ∈ RM×D,
+(3)
+where Bi-GRU(·) denotes the Bi-GRU layers, [; ]
+is the concatenate operation, and ⊙ is the element-
+wise multiplication. The output F ∈ {F a} ∪
+{F s,k}K
+k=1 encodes visual features with query-
+guided attention.
+3.3
+Multi-Modal Reasoning Strategy
+Note that the query-irreverent new boundary
+frames encoded in the anchor video feature F a has
+insufficient query-guided visual information for lat-
+ter boundary prediction. To address this issue, we
+propose a new multi-modal reasoning strategy to
+enrich the query-related knowledge in anchor fea-
+tures F a referring to the contextual information in
+siamese features {F s,k}K
+k=1. In detail, the multi-
+modal reasoning strategy consists of two compo-
+nents: a siamese knowledge aggregation module
+and a siamese knowledge reasoning module.
+Siamese knowledge aggregation Intuitively, fea-
+tures with close visual-query correlation are ex-
+pected to generate more consistent predictions of
+segment probabilities. To this end, we utilize a
+siamese knowledge aggregation module to generate
+interdependent knowledge from siamese features
+to anchor ones to enrich the contexts of anchor
+features and refine the prediction.
+We propose to propagate and integrate knowl-
+edge between the query-guided visual features F a
+and {F s,k}K
+k=1. Specifically, we first obtain their
+semantic similarities by calculating their pairwise
+cosine similarity scores as:
+C(i, k) =
+(F a
+i )(F s,k
+i
+)⊤
+∥ F a
+i ∥2∥ F s,k
+i
+∥2
+,
+(4)
+where C ∈ RM×K is interdependent similarity
+matrix, ∥ · ∥2 is l2-norm, i ∈ {1, 2, ..., M} is
+the indices of features and k ∈ {1, 2, ..., K} is the
+indices of siamese videos. Here, each anchor frame
+is needed to be enriched by only its siamese frames.
+We employ a softmax function to each row of the
+similarity matrix C as:
+C(i, k) =
+exp(C(i, k))
+� exp(C(i, k)),
+(5)
+where the new C indicates the contextual affinities
+between each anchor feature and its corresponding
+siamese features.
+Siamese knowledge reasoning
+After that, we
+propose to adaptively propagate and merge the
+siamese knowledge into the anchor features for
+enriching the query-aware information. This is es-
+pecially helpful when we determine more accurate
+boundaries for the downsampled video. Specifi-
+cally, The integration process can be formulated as:
+�F a =
+K
+�
+k=1
+C(:, k) · (F s,kW1) ∈ RM×D,
+(6)
+where �F a is the propagated semantic vector in an-
+chor video. In order to avoid over propagation and
+involves in irrelevant noisy information, we further
+exploit a residual design with a learnable weight to
+enrich the anchor video as:
+�F a = α
+K
+�
+k=1
+C(:, k) · (F s,kW1) + (1 − α)F aW2,
+(7)
+where W1, W2 ∈ RD×D are projection matrices,
+weighting factor α ∈ [0, 1] is a hyper-parameter.
+With the above formulations, the knowledge of
+the siamese samples within the same video can be
+propagated and integrated to the anchor one.
+3.4
+Grounding Heads with Soft Label
+For the final segment boundary prediction, we first
+follow the span predictor in (Zhang et al., 2020a) to
+utilize two stacked-LSTM with two corresponding
+feed-forward layers to predict the start/end scores
+
+of each frame. In details, we send the contextual
+multi-modal feature �F a ∈ RM×D into this span
+predictor and apply the softmax function on its
+two outputs to produce the probability distributions
+Ps, Pe ∈ RM of start and end boundaries. We
+utilize the rounded boundary ˆτs(e) to generate the
+coarse label vectors Ys(e) to supervise Ps, Pe as:
+L1 = fCE(Ps, Ys) + fCE(Pe, Ye),
+(8)
+where fCE represents cross-entropy loss function.
+The predicted timestamps ( ˆτs′, ˆτe′) are obtained
+from the maximum scores of start and end predic-
+tions Ps(e) of frames as:
+( ˆτs′, ˆτe′) = arg max
+ˆτs′, ˆτe′ Ps( ˆτs′)Pe( ˆτe′),
+(9)
+where 0 ≤ ˆτ ′
+s ≤ ˆτ ′
+e ≤ M.
+Since the above predictions are coarse on the seg-
+ment boundaries with boundary-bias, we further
+utilize a parallel prediction head on �F a to predict
+more fine-grained float boundaries on the down-
+sampled boundary frames. Specifically, we utilize
+the float boundary ˜τs(e) to generate the soft labels
+Y ′
+s(e), and �F a is fed into a single feed-forward layer
+to predict the float boundaries Os(e) supervised by
+our designed soft labels Y ′
+s(e) as follows:
+L2 = R1(Os(e) − Y ′
+s(e)),
+(10)
+where R1 is the smooth L1 loss. The final predicted
+segment is calculated by:
+(˜τ ′
+s, ˜τ ′
+e) = (ˆτ ′
+s+1−Os(ˆτ ′
+s), ˆτ ′
+e−1+Os(ˆτ ′
+e)). (11)
+4
+Experiments
+4.1
+Datasets and Evaluation
+ActivityNet Captions This dataset (Krishna et al.,
+2017) contains 20000 untrimmed videos from
+YouTube with 100000 textual descriptions. The
+videos are 2 minutes on average, and the annotated
+video clips have significant variation of length,
+ranging from several seconds to over 3 minutes.
+Following public split, we use 37417, 17505, and
+17031 sentence-video pairs for training, validation,
+and testing.
+TACoS
+TACoS (Regneri et al., 2013) contains
+127 videos. The videos from TACoS are collected
+from cooking scenarios, thus lacking the diversity.
+They are around 7 minutes on average. We use
+the same split as (Gao et al., 2017), which includes
+Method
+Feature
+R@1,
+R@1,
+R@5,
+R@5
+IoU=0.5 IoU=0.7 IoU=0.5 IoU=0.7
+TGN
+C3D
+28.47
+-
+43.33
+-
+CTRL
+C3D
+29.01
+10.34
+59.17
+37.54
+QSPN
+C3D
+33.26
+13.43
+62.39
+40.78
+CBP
+C3D
+35.76
+17.80
+65.89
+46.20
+GDP
+C3D
+39.27
+-
+-
+-
+VSLNet
+C3D
+43.22
+26.16
+-
+-
+CMIN
+C3D
+43.40
+23.88
+67.95
+50.73
+DRN
+C3D
+45.45
+24.36
+77.97
+50.30
+2DTAN
+C3D
+44.51
+26.54
+77.13
+61.96
+APGN
+C3D
+48.92
+28.64
+78.87
+63.19
+MGSL
+C3D
+51.87
+31.42
+82.60
+66.71
+SSRN
+C3D
+54.49
+33.15
+84.72
+68.48
+Table 1: Performance compared with the state-of-the-
+art TSG models on ActivityNet Captions dataset.
+Method
+Feature
+R@1,
+R@1,
+R@5,
+R@5,
+IoU=0.3 IoU=0.5 IoU=0.3 IoU=0.5
+TGN
+C3D
+21.77
+18.90
+39.06
+31.02
+CTRL
+C3D
+18.32
+13.30
+36.69
+25.42
+QSPN
+C3D
+20.15
+15.23
+36.72
+25.30
+CBP
+C3D
+27.31
+24.79
+43.64
+37.40
+GDP
+C3D
+24.14
+-
+-
+-
+VSLNet
+C3D
+29.61
+24.27
+-
+-
+CMIN
+C3D
+24.64
+18.05
+38.46
+27.02
+DRN
+C3D
+-
+23.17
+-
+33.36
+2DTAN
+C3D
+37.29
+25.32
+57.81
+45.04
+APGN
+C3D
+40.47
+27.86
+59.98
+47.12
+MGSL
+C3D
+42.54
+32.27
+63.39
+50.13
+SSRN
+C3D
+45.10
+34.33
+65.26
+51.85
+Table 2: Performance compared with the state-of-the-
+art TSG models on TACoS datasets.
+Method Feature
+R@1,
+R@1,
+R@5,
+R@5,
+IoU=0.5 IoU=0.7 IoU=0.5 IoU=0.7
+2DTAN
+VGG
+39.81
+23.25
+79.33
+51.15
+APGN
+VGG
+44.23
+25.64
+89.51
+57.87
+SSRN
+VGG
+46.72
+27.98
+91.37
+59.64
+CTRL
+C3D
+23.63
+8.89
+58.92
+29.57
+QSPN
+C3D
+35.60
+15.80
+79.40
+45.40
+CBP
+C3D
+36.80
+18.87
+70.94
+50.19
+GDP
+C3D
+39.47
+18.49
+-
+-
+APGN
+C3D
+48.20
+29.37
+89.05
+58.49
+SSRN
+C3D
+50.39
+31.42
+90.68
+59.94
+DRN
+I3D
+53.09
+31.75
+89.06
+60.05
+APGN
+I3D
+62.58
+38.86
+91.24
+62.11
+MGSL
+I3D
+63.98
+41.03
+93.21
+63.85
+SSRN
+I3D
+65.59
+42.65
+94.76
+65.48
+Table 3: Performance compared with the state-of-the-
+art TSG models on Charades-STA datasets.
+10146, 4589, 4083 query-segment pairs for training,
+validation and testing.
+Charades-STA Charades-STA is built on the Cha-
+rades dataset (Sigurdsson et al., 2016), which fo-
+cuses on indoor activities. The video length of
+Charades-STA dataset is 30 seconds on average,
+
+CTRL TGN 2DTAN CMIN DRN APGN SSRN
+VPS ↑
+0.45
+1.09
+1.75
+81.29 133.38 146.67 158.12
+Para. ↓
+22
+166
+363
+78
+214
+91
+184
+Table 4: Efficiency comparison in terms of video per
+second (VPS) and parameters (Para.).
+and there are 12408 and 3720 moment-query pairs
+in the training and testing sets, respectively.
+Evaluation Following previous works (Gao et al.,
+2017; Liu et al., 2021a), we adopt “R@n, IoU=m”
+as our evaluation metrics. The “R@n, IoU=m” is
+defined as the percentage of at least one of top-n
+selected moments having IoU larger than m, which
+is the higher the better.
+4.2
+Implementation Details
+For video encoding, we apply C3D (Tran et al.,
+2015) to encode the videos on all three datasets,
+and also extract the I3D (Carreira and Zisserman,
+2017) and VGG (Simonyan and Zisserman, 2014)
+features on Charades-STA dataset for fairly com-
+paring with other methods. Following previous
+works, we set the length M of the sampled anchor
+video sequences to 200 for ActivityNet Captions
+and TACoS datasets, 64 for Charades-STA dataset,
+respectively. As for sentence encoding, we utilize
+Glove word2vec (Pennington et al., 2014) to embed
+each word to a 300-dimension feature. The hidden
+state dimensions of Bi-GRU and Bi-LSTM are set
+to 512. The number K of the sampled siamese
+frames for each anchor frame is set to 4. We train
+our model with an Adam optimizer with leaning
+rate 8 × 10−4, 3 × 10−4, 4 × 10−4 for ActivityNet
+Captions, TACoS, and Charades-STA datasets, re-
+spectively. The batch size is set to 64.
+4.3
+Comparison with State-of-the-Arts
+Compared methods We compare our SSRN with
+state-of-the-art methods, including: (1) propose-
+and-rank methods: TGN (Chen et al., 2018), CTRL
+(Gao et al., 2017), QSPN (Xu et al., 2019), CBP
+(Wang et al., 2020), CMIN (Zhang et al., 2019b),
+2DTAN (Zhang et al., 2020b), APGN (Liu et al.,
+2021a), MGSL (Liu et al., 2022b). (2) proposal-
+free methods: GDP (Chen et al., 2020), VSLNet
+(Zhang et al., 2020a), DRN (Zeng et al., 2020).
+Quantitative comparison As shown in Table 1, 2
+and 3, our SSRN outperforms all the existing meth-
+ods by a large margin. Specifically, on ActivityNet
+Captions dataset, compared to the previous best
+method MGSL, we outperform it by 2.62%, 1.73%,
+Model Anchor Siamese SKA SKR SL
+R@1,
+R@1,
+IoU=0.5 IoU=0.7
+x
+✓
+×
+×
+×
+×
+42.78
+26.35
+y
+✓
+×
+×
+×
+✓
+43.64
+26.81
+z
+✓
+✓
+×
+×
+×
+45.50
+27.93
+{
+✓
+✓
+×
+✓
+×
+48.97
+29.36
+|
+✓
+✓
+✓
+✓
+×
+51.26
+31.02
+}
+✓
+✓
+✓
+✓
+✓
+54.49
+33.15
+Table 5: Main ablation studies on ActivityNet Captions
+dataset, where “Anchor" and “Siamese" denote the an-
+chor and siamese frames, “SKA" and “SKR" denote
+the siamese knowledge aggregation and siamese knowl-
+edge reasoning, “SL" denotes the usage of soft label.
+2.12%, 1.77% in all metrics, respectively.
+Al-
+though TACoS dataset suffers from similar kitchen
+background and cooking objects among the videos,
+it is worth noting that our SSRN still achieves sig-
+nificant improvements. Compared to the previ-
+ous best method MGSL, our method brings sig-
+nificant improvement of 2.06% and 1.72% in the
+strict “R@1, IoU=0.5” and “R@5, IoU=0.5” met-
+rics, respectively. On Charades-STA dataset, for
+fair comparisons with other methods, we perform
+experiments with same features (i.e., VGG, C3D,
+and I3D) reported in their papers. It shows that our
+SSRN reaches the highest results over all metrics.
+Efficiency comparison To compare the efficiency
+of our SSRN with previous methods, we make a
+fair comparison on a single Nvidia TITAN XP GPU
+on the TACoS dataset. As shown in Table 4, it can
+be observed that we achieve much faster processing
+speeds with a competitive model sizes.
+4.4
+Ablation Study
+Effect of the siamese learning strategy
+As
+shown in Table 5, we set the network without both
+siamese sampling/reasoning and soft label train-
+ing as the baseline (model x). Compared with
+the baseline, the model z additionally extracts
+siamese frames for contextual learning, and can
+apparently improve the accuracy. It directly utilizes
+average operation to aggregate siamese knowledge
+and exploit concatenation for knowledge reasoning,
+which validates that multiple frames from same
+videos can really bring some strong knowledge
+to enhance the network. When further applying
+the SKR module on model z, the model { per-
+forms better, demonstrating the effectiveness of our
+SKR module. When we further add the SKG mod-
+ule, our model | can reach a higher performance,
+which can demonstrate the effectiveness of building
+
+Anchor 
+Frames
+sampling
+Sentence Query: The person turns around to look out of a window.
+Ground Truth
+3.4s
+11.7s
+|
+|
+Sentence Query: The person turns around to look out of a window.
+Ground Truth
+3.4s
+11.7s
+|
+|
+GT
+3.4s
+11.7s
+|
+|
+GT
+3.4s
+11.7s
+|
+|
+VSLNet
+4.8s
+9.9s
+|
+|
+Ours
+3.6s
+11.6s
+|
+|
+Siamese Frames
+left
+right
+Query-related
+“turns around”
+“look”
+enrich
+enrich
+Anchor 
+Frames
+sampling
+Sentence Query: The person turns around to look out of a window.
+Ground Truth
+3.4s
+11.7s
+|
+|
+GT
+3.4s
+11.7s
+|
+|
+VSLNet
+4.8s
+9.9s
+|
+|
+Ours
+3.6s
+11.6s
+|
+|
+Siamese Frames
+left
+right
+Query-related
+“turns around”
+“look”
+enrich
+enrich
+Anchor 
+Frames
+sampling
+Sentence Query: A person is snuggling with a pillow on a chair.
+Ground Truth
+4.9s
+18.9s
+|
+|
+GT
+4.9s
+18.9s
+|
+|
+GT
+4.9s
+18.9s
+|
+|
+VSLNet
+3.7s
+15.6s
+|
+|
+Ours
+4.9s
+18.4s
+|
+|
+Siamese Frames
+left
+right
+Query-related
+“pillow”
+“snuggling”
+enrich
+enrich
+Figure 3: The visualization examples to show the benefits from the siamese frames. Due to the boundary-bias
+during the sparse sampling process, previous VSLNet method filters out the true-positive boundary frames and
+fails to predict the accurate boundaries. Instead, our siamese learning strategy supplements the query-related
+information of the adjacent frames into the ambiguous downsampled boundary-frames for predicting more precise
+boundaries.
+Number
+K=1
+K=2
+K=4
+K=8
+R@1, IoU=0.5
+50.45
+52.10
+54.49
+54.62
+R@1, IoU=0.7
+29.64
+30.78
+33.15
+33.27
+Table 6: The effect of the number K of the sampled
+siamese frames on ActivityNet Captions dataset.
+the interdependent knowledge (i.e., siamese knowl-
+edge) for integrating the samples. It can also prove
+that adaptively reasoning by our siamese knowl-
+edge is better than the purely average operation. We
+think that the siamese knowledge not only serves
+as the knowledge-routed representation, but also
+implicitly constrains the semantic consistency of
+frames in the space of frame-text features.
+Effect of the usage of soft label
+We also inves-
+tigate whether our soft label (float value) of the
+segment boundary contributes to the performance
+of our model. As shown in Table 5, directly apply-
+ing the soft label learning to the baseline does not
+bring significant performance improvement (model
+y). This is mainly because that the boundary frame
+may be query-irrelevant and its feature is not able
+to be accurately matched with the query. Instead,
+comparing model } with model |, model } en-
+riches the boundary frames with siamese contexts
+and supplements them with the neighboring query-
+related visual information. Therefore, it brings
+significant improvement by using the soft label in
+training process.
+Effect of the number of siamese frames
+We
+compare our method with various number of
+siamese frames as shown in Table 6. When adding
+the siamese sample number K from 1 to 8, our
+method dynamically promotes the accuracy. Such
+improvement can demonstrate that more siamese
+samples can bring richer knowledge, which makes
+our network benefited from it. Although the ac-
+curacy is increasing with the number of siamese
+Methods
+Variant
+R@1,
+R@1,
+IoU=0.5 IoU=0.7
+VSLNet
+Origin
+43.22
+26.16
++siamese
+50.38
+30.06
+CBLN
+Origin
+48.12
+27.60
++siamese
+56.86
+30.79
+MGSL
+Origin
+51.87
+31.42
++siamese
+58.77
+33.41
+Table 7: We apply our siamese learning strategy to ex-
+isting TSG models on ActivityNet Captions dataset.
+frames, we observe that the improvement from the
+number 4 to 8 is slight. We think the reason is
+the saturation of knowledge, i.e., the model has
+enough knowledge to learn the task on this dataset.
+Hence, it is almost meaningless to purely increase
+the siamese frames. To balance the training time
+and accuracy, we assign K = 4 in our final version.
+Plug-and-Play
+Our proposed siamese learning
+strategy is flexible and can be adopted to other
+TSG methods for anchor feature enhancement. As
+shown in Table 7, we directly apply siamese learn-
+ing strategy into existing module for anchor feature
+enriching without using soft label training. It shows
+that our siamese learning strategy can provide more
+contextual and fine-grained information for anchor
+feature encoding, bringing large improvement.
+4.5
+Qualitative Results
+In Figure 3, we show two visualization examples to
+qualitatively analyze what kind of knowledge does
+the siamese frames bring to the anchor frames. It
+is unavoidable to lose some visual contents when
+sparsely sampling from the video. Especially for
+the boundary frames that are easily to be filtered
+out by sampling, the visual content of the newly
+sampled boundary may lose query-relevant infor-
+mation (e.g., brown words in figure). However, we
+can obtain the absent contents from their siamese
+
+frames due to different sampling indices and du-
+ration. Hence, our siamese frames can enrich and
+supplement the sampled frames with more con-
+secutive query-related visual semantics to make a
+fine-grained video comprehension, keeping the ap-
+propriate segment length of the sampled video for
+more accurate boundary prediction.
+5
+Conclusion
+In this paper, we propose a novel Siamese Sampling
+and Reasoning Network (SSRN) to alleviate the
+limitations of both boundary-bias and reasoning-
+bias in existing TSG methods. In addition to the
+original anchor frames, our model also samples a
+certain number of siamese frames from the same
+video to enrich and refine the visual semantics of
+the anchor frames. A soft label is further exploited
+to supervise the enhanced anchor features for pre-
+dicting more accurate segment boundaries. Exper-
+imental results show both effectiveness and effi-
+ciency of our SSRN on three challenging datasets.
+Limitations
+This work analyzes an interesting problem of how
+to learn from inside to address the limitation of the
+boundary-bias on the temporal sentence ground-
+ing. Since our method targets on the issue of long
+video sampling, it may be not helpful to handle the
+short video processing but still can improve the con-
+textual representation learning for the short video.
+Besides, our sampled siamese frames would bring
+extra burden (e.g., computation, memory and pa-
+rameters) during the training and testing. Therefore,
+a more light way to ease the siamese knowledge
+extraction is a promising future direction.
+6
+Acknowledgments
+This work was supported by National Natu-
+ral Science Foundation of China (No.61972448,
+No.62272328, No.62172038 and No.62172068).
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+

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+arXiv:2301.00035v1  [math.QA]  30 Dec 2022
+Guay’s affine Yangians and non-rectangular W-algebras
+Mamoru Ueda
+Abstract
+We construct a non-trivial homomorphism from the Guay’s affine Yangian to the univer-
+sal enveloping algebra of non-rectangular W -algebras of type A. In order to construct the
+homomorphism, we extend the Guay’s affine Yangian and its coproduct.
+1
+Introduction
+A W-algebra appeared in the study of two dimensional conformal field theories ([25]) and has
+been studied by wide range physicists and mathematicians such that integrable systems, and four-
+dimensional gauge theories. In this paper, we relate the W-algebras of type A to one quantum
+group, which is called the Guay’s affine Yangian.
+The Guay’s affine Yangian Yℏ,ε(�sl(n)) ([11] and [12]) is a 2-parameter Yangian and is the
+deformation of the universal enveloping algebra of the central extension of sl(n)[u±1, v].
+The
+Guay’s affine Yangian Yℏ,ε(�sl(n)) has an evaluation map ([12] and [18])
+evx : Yℏ,ε(�sl(n)) → the standard degreewise completion of U(�gl(n))
+and a coproduct ([12] and [13])
+∆a,b : Yℏ,ε(�sl(n)) → Yℏ,ε(�sl(n))�⊗Yℏ,ε(�sl(n)),
+where Yℏ,ε(�sl(n))�⊗Yℏ,ε(�sl(n)) is the standard degreewise completion of Yℏ,ε(�sl(n))⊗2.
+Let us take an integer a ≥ n. We extend the Guay’s affine Yangian Yℏ,ε(�sl(n)) to the new
+associative algebra Y a
+ℏ,ε(�sl(n)). One of the features of Y a
+ℏ,ε(�sl(n)) is that there exist the following
+natural algebra homomorphisms
+Ψ1 : Yℏ,ε(�sl(n)) → Y a
+ℏ,ε(�sl(n)),
+Ψ2 : U(�gl(a)) → Y a
+ℏ,ε(�sl(n))
+and Y a
+ℏ,ε(�sl(n)) is generated by the image of Ψ1 and Ψ2.
+The algebra Y a
+ℏ,ε(�sl(n)) has a map
+corresponding to the evaluation map
+�evx : Y a
+ℏ,ε(�sl(n)) → the standard degreewise completion of U(�gl(a)).
+For a ≥ b ≥ n, there also exists a map corresponding to the coproduct
+∆a,b : Y b
+ℏ,ε(�sl(n)) → Y a
+ℏ,ε−(a−b)ℏ(�sl(n))�⊗Y b
+ℏ,ε(�sl(n))/ ∼,
+where Y a
+ℏ,ε−(a−b)ℏ(�sl(n))�⊗Y b
+ℏ,ε(�sl(n))/ ∼ is the standard degreewise completion of the tensor alge-
+bra Y a
+ℏ,ε−(a−b)ℏ(�sl(n)) ⊗ Y b
+ℏ,ε(�sl(n)) divided by one relation.
+A W-algebra Wk(g, f) is a vertex algebra associated with a finite dimensional reductive Lie
+algebra g and its nilpotent element f. It is defined by the quantized Drinfeld-Sokolov reduction
+([16] and [7]). In this paper, we consider the case that g = gl(N) and its nilpotent element whose
+Jordan block is of type (1q1−q2, 2q2−q3, 3q3−q4, · · · , (l − 1)ql−1−ql, lql), where
+N = q1 + q2 + · · · + ql,
+q1 ≥ q2 ≥ · · · ≥ ql.
+1
+
+In this case, there exists an injective homomorphism called Miura map (see [16])
+µ: Wk(g, f) → ⊗l
+i=1V κi(gl(qi)),
+where V κi(gl(qi)) is the universal affine vertex algebra associated with gl(qi) and its inner product
+κi. Taking the universal enveloping algebra of both sides in the sense of [8] and [20], we obtain an
+injective homomorphism
+�µ: Wk(g, f) → U(�gl(q1))�⊗ · · · �⊗U(�gl(ql)),
+where U(�gl(q1))�⊗ · · · �⊗U(�gl(ql)) is the standard degreewise completion of ⊗l
+i=1U(gl(qi)).
+In the case that q1 = q2 = · · · = ql = n, the W-algebra Wk(g, f) is called the rectangular
+W-algebra of type A. In this case, by a direct computation, the author [24] have constructed a
+surjective homomorphism
+�Φ: Y b
+ℏ,ε(�sl(n)) → U(Wk(gl(ln), f)),
+where U(Wk(gl(ln), f)) is the universal enveloping algebra of Wk(gl(ln), f). In [19], Kodera and
+the author showed that this homomorphism can be written down by using the coproduct ([12] and
+[13]) and evaluation map ([12] and [18]) for the Guay’s affine Yangian as follows;
+(ev0 ⊗ evℏα ⊗ · · · ⊗ evℏ(l−1)α) ◦ ∆ ⊗ id⊗l−1) ◦ · · · ◦ (∆ ⊗ id) ◦ ∆ = �µ ◦ �Φ,
+where α = k + (l − 1)n.
+We extend this result to the general nilpotent element. In finite setting, Brundan-Kleshchev
+[3] gave a surjective homomorphism from a shifted Yangian, which is a subalgebra of the finite
+Yangian associated with gl(n), to a finite W-algebra ([21]) of type A for its general nilpotent
+element. A finite W-algebra Wfin(g, f) is an associative algebra associated with a reductive Lie
+algebra g and its nilpotent element f and is a finite analogue of a W-algebra Wk(g, f) ([5] and
+[1]). In [6], De Sole, Kac and Valeri constructed a homomorphism from the finite Yangian of
+type A to the finite W-algebras of type A by using the Lax operator, which is a restriction of the
+homomorphism given by Brundan-Kleshchev in [3].
+Motivated by the work of De Sole, Kac and Valeri [6], by a direct computation, the author
+[23] constructed a surjective homomorphism from the Guay’s affine Yangian to the universal
+enveloping algebra of In this article, we constructed a homomorphism from the Guay’s affine
+Yangian Yℏ,ε(�sl(ql)) to the universal enveloping algebra of the W-algebra Wk(gl(N), f).
+Theorem 1.1. Let ql ≥ 3.
+We assume that ε
+ℏ = −(k + N).
+Then, there exists an algebra
+homomorphism
+Φ: Yℏ,ε(�sl(ql)) → U(Wk(gl(N), f))
+satisfying
+l
+�
+i=1
+�evai ◦ ∆q1,q2 ⊗ id⊗l−1) ◦ · · · ◦ (∆ql−2,ql−1 ⊗ id) ◦ ∆ql−1,ql ◦ Ψ1 = �µ ◦ Φ,
+where ai = −ℏ
+l�
+y=i+1
+(k + N − qi) and εi = ε − (qi − ql)ℏ.
+We hope that this theorem will help to resolve the genralized AGT (Alday-Gaiotto-Tachikawa)
+conjecture. The AGT conjecture suggests that there exists a representation of the principal W-
+algebra of type A on the equivariant homology space of the moduli space of U(r)-instantons.
+Schiffmann and Vasserot [22] gave this representation by using an action of the Yangian associated
+with �gl(1) on this equivariant homology space. It is conjectured in [4] that an action of an iterated
+W-algebra of type A on the equivariant homology space of the affine Laumon space will be given
+through an action of an affine shifted Yangian constructed in [8].
+2
+
+2
+Guay’s affine Yangian
+Let us recall the definition of the Guay’s affine Yangian. The Guay’s affine Yangian Yε1,ε2(�sl(n))
+was first introduced by Guay ([11] and [12]) and is the deformation of the universal enveloping
+algebra of the central extension of sl(n)[u±1, v].
+Definition 2.1. Let n ≥ 3 and an n × n matrix (ai,j)1≤i,j≤n be
+aij =
+
+
+
+
+
+
+
+
+
+2
+if i = j,
+−1
+if j = i ± 1,
+1
+if (i, j) = (0, n − 1), (n − 1, 0),
+0
+otherwise.
+The Guay’s affine Yangian Yℏ,ε(�sl(n)) is the associative algebra generated by X+
+i,r, X−
+i,r, Hi,r (i ∈
+{0, 1, · · · , n − 1}, r = 0, 1) subject to the following defining relations:
+[Hi,r, Hj,s] = 0,
+(2.2)
+[X+
+i,0, X−
+j,0] = δijHi,0,
+(2.3)
+[X+
+i,1, X−
+j,0] = δijHi,1 = [X+
+i,0, X−
+j,1],
+(2.4)
+[Hi,0, X±
+j,r] = ±aijX±
+j,r,
+(2.5)
+[ ˜Hi,1, X±
+j,0] = ±aij
+�
+X±
+j,1
+�
+if (i, j) ̸= (0, n − 1), (n − 1, 0),
+(2.6)
+[ ˜H0,1, X±
+n−1,0] = ∓
+�
+X±
+n−1,1 − (ε + n
+2 ℏ)X±
+n−1,0
+�
+,
+(2.7)
+[ ˜Hn−1,1, X±
+0,0] = ∓
+�
+X±
+0,1 + (ε + n
+2 ℏ)X±
+0,0
+�
+,
+(2.8)
+[X±
+i,1, X±
+j,0] − [X±
+i,0, X±
+j,1] = ±aij
+ℏ
+2{X±
+i,0, X±
+j,0} if (i, j) ̸= (0, n − 1), (n − 1, 0),
+(2.9)
+[X±
+0,1, X±
+n−1,0] − [X±
+0,0, X±
+n−1,1] = ±ℏ
+2{X±
+0,0, X±
+n−1,0} − (ε + n
+2 ℏ)[X±
+0,0, X±
+n−1,0],
+(2.10)
+(ad X±
+i,0)1−aij(X±
+j,0) = 0 if i ̸= j,
+(2.11)
+where ˜Hi,1 = Hi,1 − ℏ
+2H2
+i,0.
+Remark 2.12. The defining relations of Yℏ,ε(�sl(n)) are different from those of Yε1,ε2(�sl(n)) which
+is called the Guay’s affine Yangian in [18]. In [18], generators of Yε1,ε2(�sl(n)) are denoted by
+{x±
+i,r, hi,r | 0 ≤ i ≤ n − 1, r ∈ Z≥0}
+with 2-parameters ε1 and ε2. Actually, the algebra Yℏ,ε(�sl(n)) is isomorphic to Yε1,ε2(�sl(n)). The
+isomorphism Ψ from Yℏ,ε(�sl(n)) to Yε1,ε2(�sl(n)) is given by
+Ψ(Hi,0) = hi,0,
+Ψ(X±
+i,0) = x±
+i,0,
+Ψ(Hi,1) =
+
+
+
+h0,1
+if i = 0,
+hi,1 + i
+2(ε1 − ε2)hi,0
+if i ̸= 0,
+ℏ = ε1 + ε2,
+ε = −nε2.
+In this paper, we do not use Ya,b(�sl(n)) in the meaning of [18].
+Let us recall the evaluation map for the Guay’s affine Yangian (see [12] and [18]). We set a
+Lie algebra
+�gl(n)c =
+� �
+s∈Z
+�
+1≤i,j≤n
+Ei,jts�
+⊕ Cc ⊕ Cz
+3
+
+whose commutator relations are determined by
+[Ep,qts, Ei,jtu] = δi,qEp,jts+u − δp,jEi,qts+u + sδi,qδp,jδs+u,0c + sδp,qδi,jδs+u,0z,
+c and z are central elements.
+Next, we introduce a completion of U(�gl(n)c)/U(�gl(n)c)(z − 1) following [20]. We set the grading
+of U(�gl(n))/U(�gl(n)c)(z − 1) as deg(Ei,jts) = s and deg(c) = 0. Then, U(�gl(n)c)/U(�gl(n)c)(z − 1)
+becomes a graded algebra and we denote the set of the degree d elements of U(�gl(n)c)/U(�gl(n)c)(z−
+1) by U(�gl(n)c)d. We obtain the completion
+U(�gl(n)c)comp =
+�
+d∈Z
+U(�gl(n)c)comp,d,
+where
+U(�gl(n))c
+comp,d = lim
+←−
+N
+U(�gl(n)c)d/
+�
+r>N
+U(�gl(n)c)d−rU(�gl(n)c)r.
+The evaluation map for the Guay’s affine Yangian is a non-trivial homomorphism from the Guay’s
+affine Yangian to U(�gl(n)c)comp. Here after, we denote
+hi =
+�
+En,n − E1,1 + c
+(i = 0),
+Eii − Ei+1,i+1
+(1 ≤ i ≤ n − 1),
+x+
+i =
+�
+En,1t
+(i = 0),
+Ei,i+1
+(1 ≤ i ≤ n − 1),
+x−
+i =
+�
+E1,nt−1
+(i = 0),
+Ei+1,i
+(1 ≤ i ≤ n − 1).
+Theorem 2.13 (Section 6 in [12] and Theorem 3.8 in [18]). Set c = −nℏ − ε
+ℏ
+. Then, there exists
+an algebra homomorphism
+evℏ,ε : Yℏ,ε(�sl(n)) → U(�gl(n)c)comp
+uniquely determined by
+evℏ,ε(X+
+i,0) = x+
+i ,
+evℏ,ε(X−
+i,0) = x−
+i ,
+evℏ,ε(Hi,0) = hi,
+evℏ,ε(Hi,1) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏch0 − ℏEn,n(E1,1 − c)
++ℏ
+�
+s≥0
+n
+�
+k=1
+En,kt−sEk,nts − ℏ
+�
+s≥0
+n
+�
+k=1
+E1,kt−s−1Ek,1ts+1
+if i = 0,
+− i
+2ℏhi − ℏEi,iEi+1,i+1
++ℏ
+�
+s≥0
+i
+�
+k=1
+Ei,kt−sEk,its + ℏ
+�
+s≥0
+n
+�
+k=i+1
+Ei,kt−s−1Ek,its+1
+−ℏ
+�
+s≥0
+i
+�
+k=1
+Ei+1,kt−sEk,i+1ts − ℏ
+�
+s≥0
+n
+�
+k=i+1
+Ei+1,kt−s−1Ek,i+1ts+1
+if i ̸= 0,
+evℏ,ε(X+
+i,1) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏcx+
+0 + ℏ
+�
+s≥0
+n
+�
+k=1
+En,kt−sEk,1ts+1
+if i = 0,
+− i
+2ℏx+
+i + ℏ
+�
+s≥0
+i
+�
+k=1
+Ei,kt−sEk,i+1ts + ℏ
+�
+s≥0
+n
+�
+k=i+1
+Ei,kt−s−1Ek,i+1ts+1
+if i ̸= 0,
+4
+
+evℏ,ε(X−
+i,1) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏcx−
+0 + ℏ
+�
+s≥0
+n
+�
+k=1
+E1,kt−s−1Ek,nts,
+if i = 0,
+− i
+2ℏx−
+i + ℏ
+�
+s≥0
+i
+�
+k=1
+Ei+1,kt−sEk,its + ℏ
+�
+s≥0
+n
+�
+k=i+1
+Ei+1,kt−s−1Ek,its+1
+if i ̸= 0.
+We recall the coproduct for the Guay’s affine Yangian. Let ∆+ (resp. ∆+
+re) be the set of
+positive roots (resp.
+positive real roots) of �sl(n).
+We denote the multiplicity of a root γ by
+p(γ). We take root vectors {x(r)
+±γ | 1 ≤ r ≤ p(γ)} for γ ∈ ∆+ satisfying (x(r)
+γ , x(s)
+−γ) = δr,s, where
+( , ) is the standard invariant symmetric bilinear form. We denote the simple roots of �sl(n) by
+{αi | 0 ≤ i ≤ n − 1}.
+By Theorem 6.1 in [12] and Theorem 6.9 in [14], we have an embedding ξ from U(�sl(n)) ⊂
+U(�gl(n))c to Yℏ,ε(�sl(n)) determined by
+ξ(hi) = Hi,0,
+ξ(x±
+i ) = X±
+i,0.
+We identify U(�sl(n)) and its image via ξ.
+We set the degree of the Guay’s affine Yangian as follows;
+deg(Hi,r) = 0,
+deg(X±
+i,r) = ±δi,0.
+By this degree, we can define the standard degree completion of Yℏ,ε(�sl(n))⊗2. We denote it by
+Yℏ,ε(�sl(n))�⊗Yℏ,ε(�sl(n)).
+Theorem 2.14 (Theorem 5.2 in [13]). There exists an algebra homomorphism
+∆: Yℏ,ε(�sl(n)) → Yℏ,ε(�sl(n))�⊗Yℏ,ε(�sl(n))
+determined by
+∆(Hi,0) = Hi,0 ⊗ 1 + 1 ⊗ Hi,0,
+∆(X±
+i,0) = X±
+i,0 ⊗ 1 + 1 ⊗ X±
+i,0,
+∆(Hi,1) = Hi,1 ⊗ 1 + 1 ⊗ Hi,1
++ ℏ(Hi,0 ⊗ Hi,0 −
+�
+γ∈∆+
+re
+(αi, γ)x(1)
+−γ ⊗ x(1)
+γ ),
+∆(X+
+i,1) = X+
+i,1 ⊗ 1 + 1 ⊗ X+
+i,1
++ ℏ(Hi,0 ⊗ X+
+i,0 −
+�
+γ∈∆+
+p(γ)
+�
+r=1
+x(r)
+−γ ⊗ [x+
+i , x(r)
+γ ]),
+∆(X−
+0,1) = X−
+i,1 ⊗ 1 + 1 ⊗ X−
+i,1
++ ℏ(Hi,0 ⊗ X+
+i,0 +
+�
+γ∈∆+
+p(γ)
+�
+r=1
+[x−
+i , x(r)
+−γ] ⊗ x(r)
+γ ).
+3
+Extension to the new Yangian
+We extend the definition of the Guay’s affine Yangian.
+5
+
+Definition 3.1. Let a ≥ n. We define �Y a
+ℏ,ε(�sl(n)) by the associative algebra whose generators
+are {Hi,r, X±
+i,r | 0 ≤ i ≤ n − 1, r ∈ Z≥0} and {ei,jts, ca | 1 ≤ i, j ≤ n, s ∈ Z} with the following
+defining relations;
+the defining relations (2.2)-(2.11),
+[ei,jts, eu,vtw] = δj,uei,vts+w − δi,veu,jts+w + sδs+w,0δi,vδu,jca + δi,jδu,v,
+ca is a central element,
+Hi,0 =
+�
+en,n − e1,1 + ca if i = 0,
+ei,i − ei+1,i+1 if i ̸= 0,
+X+
+i,0 =
+�
+en,1t if i = 0,
+ei,i+1 if i ̸= 0,
+X−
+i,0 =
+�
+e1,nt−1 if i = 0,
+ei+1,i if i ̸= 0.
+We set the degree on �Y a
+ℏ,ε(�sl(n)) as
+deg(Hi,r) = 0, deg(X±
+i,r) = ±δi,0, deg(ei,jts) = s, deg(ca) = 0.
+We define �Y a
+ℏ,ε(�sl(n)) as the standard degreewise completion of �Y a
+ℏ,ε(�sl(n)). Let us define �evℏ,ε(Hi,1)
+as an element of �Y a
+ℏ,ε(�sl(n)) in the same formula as the one in (2.13). By a direct computation,
+we obtain
+[ �evℏ,ε(Hi,1), ev,jtw]
+= i
+2ℏδi,jej,vtw − i
+2ℏδi+1,jej,vtw + ℏδi,jev,jtwei+1,i+1 + ℏδi+1,jei,iev,jtw
+− ℏ
+�
+s≥0
+δ(j ≤ i)ei,jt−sev,its+w − ℏ
+�
+s≥0
+i
+�
+u=1
+δi,jev,utw−seu,its
+− ℏ
+�
+s≥0
+δ(j > i)ei,jt−s−1ev,its+w+1 − ℏ
+�
+s≥0
+n
+�
+u=i+1
+δi,jev,utw−s−1eu,its+1
++ ℏ
+�
+s≥0
+δ(j ≤ i)ei+1,jt−sev,i+1ts+w + ℏ
+�
+s≥0
+i
+�
+u=1
+δi+1,jev,utw−seu,i+1ts
++ ℏ
+�
+s≥0
+δ(j > i)ei+1,jt−s−1ev,i+1ts+w+1 + ℏ
+�
+s≥0
+n
+�
+u=i+1
+δi+1,jev,utw−s−1eu,i+1ts+1,
+(3.2)
+[ �evℏ,ε(Hi,1), ej,vtw]
+= − i
+2ℏδi,jej,vtw + i
+2ℏδi+1,jej,vtw − ℏδi,jej,vtwei+1,i+1 − ℏδi+1,jei,iej,vtw
++ ℏ
+�
+s≥0
+i
+�
+u=1
+δi,jei,ut−seu,vts+w + ℏ
+�
+s≥0
+δ(j ≤ i)ei,vtw−sej,its
++ ℏ
+�
+s≥0
+n
+�
+u=i+1
+δi,jei,ut−s−1eu,vts+w+1 + ℏ
+�
+s≥0
+δ(j > i)ei,vtw−s−1ej,its+1
+− ℏ
+�
+s≥0
+i
+�
+u=1
+δi+1,jei+1,ut−seu,vts+w − ℏ
+�
+s≥0
+δ(j ≤ i)ei+1,vtw−sej,i+1ts
+− ℏ
+�
+s≥0
+n
+�
+u=i+1
+δi+1,jei+1,ut−s−1eu,vts+w+1
+− ℏ
+�
+s≥0
+δ(j > i)ei+1,vtw−s−1ej,i+1ts+1
+(3.3)
+6
+
+for all i ̸= 0, 1 ≤ j ≤ n and n < v ≤ b. By a direct computation, we also obtain
+[ �evℏ,ε(H0,1), ev,jtw]
+= −ℏcaδn,jev,jtw + ℏcaδ1,jev,jtw + ℏδ1,jen,nev,jtw + δn,jℏev,jtwe1,1 − δn,jℏcaev,jtw
+− ℏ
+�
+s≥0
+en,jt−sev,nts+w − ℏ
+�
+s≥0
+n
+�
+u=1
+δn,jev,utw−seu,nts
++ ℏ
+�
+s≥0
+e1,jt−s−1ev,1tw+s+1 + ℏ
+�
+s≥0
+n
+�
+u=1
+δj,1ev,utw−s−1eu,1ts+1,
+(3.4)
+[ �evℏ,ε(H0,1), ej,vtw]
+= ℏcaδj,nej,vtw − ℏcaδ1,jej,vtw − ℏδ1,jen,nej,vtw − ℏδj,nej,vtwe1,1 + ℏδj,ncaej,vtw
++ ℏ
+�
+s≥0
+n
+�
+u=1
+δj,nen,ut−seu,vts+w + ℏ
+�
+s≥0
+en,vtw−sej,nts
+− ℏ
+�
+s≥0
+n
+�
+u=1
+δ1,je1,ut−s−1eu,vtw+s+1 − ℏ
+�
+s≥0
+e1,vtw−s−1ej,1ts+1
+(3.5)
+for all 1 ≤ j ≤ n and n < v ≤ b.
+We set an associative algebra �Y a
+ℏ,ε(�sl(n)) is a quotient algebra divided by
+[Hi,1, ev,jtw] = [ �evℏ,ε(Hi,1), ev,jtw],
+(3.6)
+[Hi,1, ej,vtw] = [ �evℏ,ε(Hi,1), ej,vtw],
+(3.7)
+[Hi−1,1, ev,itw] + [Hi,1, ev,itw] = [ �evℏ,ε(Hi−1,1), ev,itw] + [ �evℏ,ε(Hi,1), ev,itw],
+(3.8)
+[Hi−1,1, ev,itw] + [Hi,1, ev,itw] = [ �evℏ,ε(Hi−1,1), ev,itw] + [ �evℏ,ε(Hi,1), ev,itw],
+(3.9)
+[H0,1, ev,jtw] = [ �evℏ,ε(H0,1), ev,jtw],
+(3.10)
+[H0,1, ej,vtw] = [ �evℏ,ε(H0,1), ej,vtw],
+(3.11)
+[H0,1, ev,ntw] + [Hn−1,1, ev,ntw] = [ �evℏ,ε(H0,1), ev,ntw] + [ �evℏ,ε(Hn−1,1), ev,ntw],
+(3.12)
+[H0,1, ev,1tw] + [H1,1, ev,1tw] = [ �evℏ,ε(H1,1), ev,1tw] + [ �evℏ,ε(H1,1), ev,1tw],
+(3.13)
+[H0,1, en,vtw] + [Hn−1,1, en,vtw] = [ �evℏ,ε(H0,1), en,vtw] + [ �evℏ,ε(Hn−1,1), en,vtw],
+(3.14)
+[H0,1, e1,vtw] + [H1,1, e1,vtw] = [ �evℏ,ε(H1,1), e1,vtw] + [ �evℏ,ε(H1,1), e1,vtw].
+(3.15)
+By the definition of Y a
+ℏ,ε(�sl(n)), we have two homomorphisms;
+Ψ1 : Yℏ,ε(�sl(n)) → Y a
+ℏ,ε(�sl(n))
+determined by
+Ψ1(Hi,r) = Hi,r,
+Ψ1(X±
+i,r) = X±
+i,r
+and
+Ψ2 : U(�gl(n)ca) → Y a
+ℏ,ε(�sl(n))
+determined by
+Ψ2(ei,jts) = ei,jts
+Ψ2(ca) = ca.
+By the definition of Y a
+ℏ,ε(�sl(n)), we find that we can construct a non-trivial homomorphism
+from Y a
+ℏ,ε(�sl(n)) to the standard degreewise completion of the universal enveloping algebra of �gl(a)
+as follows.
+Theorem 3.16. For x ∈ C, there exists an algebra homomorphism
+�evx
+ℏ,ε : Y a
+ℏ,ε(�sl(n)) → U(�gl(n)ca)comp
+7
+
+determined by
+�evx
+ℏ,ε(ei,jts) = ei,jts,
+�evx
+ℏ,ε(ca) = −nℏ + ε
+ℏ
+�evx
+ℏ,ε(Hi,1) = evℏ,ε(Hi,1) + xHi,0,
+�evx
+ℏ,ε(X±
+i,1) = evℏ,ε(X±
+i,1) + xX±
+i,0.
+We can construct a map corresponding to a coproduct. Let a ≥ b ≥ n. We take a degree for
+Y a
+ℏ,ε(�sl(n)) ⊗ Y b
+ℏ,ε(�sl(n)) determined by
+deg(Hi,r ⊗ 1) = deg(1 ⊗ Hi,r) = 0,
+deg(X±
+i,r ⊗ 1) = deg(1 ⊗ X±
+i,r) = ±δi,0,
+deg(ei,jts ⊗ 1) = deg(1 ⊗ ei,jts) = s,
+deg(ca ⊗ 1) = deg(1 ⊗ cb) = 0.
+We set Y a
+ℏ,ε(�sl(n))�⊗Y b
+ℏ,ε(�sl(n)) as the standard degreewise completion of Y a
+ℏ,ε(�sl(n)) ⊗ Y b
+ℏ,ε(�sl(n)).
+Moreover, we set Y a
+ℏ,ε(�sl(n))�⊗Y b
+ℏ,ε(�sl(n)) as a quotient algebra of Y a
+ℏ,ε(�sl(n))�⊗Y b
+ℏ,ε(�sl(n)) divided
+by
+ca ⊗ 1 − 1 ⊗ cb = −(a − b).
+Theorem 3.17. There exists an algebra homomorphism
+∆a,b : Y b
+ℏ,ε−(a−b)ℏ(�sl(n)) → Y a
+ℏ,ε(�sl(n))�⊗Y b
+ℏ,ε(�sl(n))
+determined by
+∆a,b(ei,jts) = ei,jts ⊗ 1 + 1 ⊗ δ(i, j ≤ b)ei,jts,
+∆a,b(Hi,0) = Hi,0 ⊗ 1 + 1 ⊗ Hi,0,
+∆a,b(X±
+i,0) = X±
+i,0 ⊗ 1 + 1 ⊗ X±
+i,0,
+∆a,b(Hi,1) =
+�
+(H0,1 + B0) ⊗ 1 + 1 ⊗ H0,1 + A0 − F0 if i = 0,
+(Hi,1 + Bi) ⊗ 1 + 1 ⊗ H0,1 + Ai − Fi if i ̸= 0,
+∆a,b(X+
+i,1) =
+�
+(X+
+0,1 + B+
+0 ) ⊗ 1 + 1 ⊗ X+
+0,1 + A+
+0 − F +
+0 if i = 0,
+(X+
+i,1 + B+
+i ) ⊗ 1 + 1 ⊗ X+
+0,1 + A+
+i − F +
+i
+if i ̸= 0,
+∆a,b(X−
+i,1) =
+�
+(X−
+0,1 + B−
+0 ) ⊗ 1 + 1 ⊗ X−
+0,1 + A−
+0 − F −
+0 if i = 0,
+(X−
+i,1 + B−
+i ) ⊗ 1 + 1 ⊗ X−
+0,1 + A−
+i − F −
+i
+if i ̸= 0,
+where
+Fi =
+
+
+
+
+
+
+
+
+
+
+
+ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,itw ⊗ ei,vt−w − ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,i+1tw ⊗ ei+1,vt−w if i ̸= 0,
+ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,ntw ⊗ en,vt−w − ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,1tw ⊗ e1,vt−w if i = 0.
+F +
+i
+=
+
+
+
+
+
+
+
+
+
+
+
+ℏ
+�
+w∈Z
+b
+�
+u=n+1
+eu,1t−w ⊗ en,utw+1 if i = 0,
+ℏ
+�
+w∈Z
+b
+�
+u=n+1
+eu,i+1t−w ⊗ ei,utw if i ̸= 0,
+8
+
+F −
+i
+=
+
+
+
+
+
+
+
+
+
+
+
+ℏ
+�
+w∈Z
+b
+�
+u=n+1
+eu,nt−w ⊗ e1,utw−1 if i = 0,
+ℏ
+�
+w∈Z
+b
+�
+u=n+1
+eu,it−w ⊗ ei+1,utw if i ̸= 0,
+Ai =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−ℏ(e1,1 ⊗ en,n + en,n ⊗ e1,1) + ℏ(en,n − e1,1) ⊗ cb + ℏca ⊗ (en,n − e1,1) + ℏca ⊗ cb
++ℏ
+�
+s≥0
+n
+�
+u=1
+(−eu,nt−s−1 ⊗ en,uts+1 + en,ut−s ⊗ eu,nts)
+−ℏ
+�
+s≥0
+n
+�
+u=1
+(−eu,1t−s ⊗ e1,uts + e1,ut−s−1 ⊗ eu,1ts+1)
+if i = 0,
+−ℏ(ei,i ⊗ ei+1,i+1 + ei+1,i+1 ⊗ ei,i)
++ℏ
+�
+s≥0
+i
+�
+u=1
+(−eu,it−s−1 ⊗ ei,uts+1 + ei,ut−s ⊗ eu,its)
++ℏ
+�
+s≥0
+n
+�
+u=i+1
+(−eu,it−s ⊗ ei,uts + ei,ut−s−1 ⊗ eu,its+1)
+−ℏ
+�
+s≥0
+i
+�
+u=1
+(−eu,i+1t−s−1 ⊗ ei+1,uts+1 + ei+1,ut−s ⊗ eu,i+1ts)
+−ℏ
+�
+s≥0
+n
+�
+u=i+1
+(−eu,i+1t−s ⊗ ei+1,uts + ei+1,ut−s−1 ⊗ eu,i+1ts+1)
+if i ̸= 0,
+A+
+i =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏca ⊗ en,1t + ℏ
+�
+s≥0
+n
+�
+u=1
+eu,1t−s ⊗ en,uts+1 − ℏ
+�
+s≥0
+n
+�
+u=1
+en,ut−s ⊗ eu,1ts+1
+if i = 0,
+ℏ
+�
+s≥0
+i
+�
+u=1
+(−eu,i+1t−s−1 ⊗ ei,uts+1 + ei,ut−s ⊗ eu,i+1ts)
++ℏ
+�
+s≥0
+n
+�
+u=i+1
+(−eu,i+1t−s ⊗ ei,uts + ei,ut−s−1 ⊗ eu,i+1ts+1)
+if i ̸= 0,
+A−
+i =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏe1,nt−1 ⊗ cb + ℏ
+�
+s≥0
+n
+�
+u=1
+(−eu,nt−s−1 ⊗ e1,uts + e1,ut−s−1 ⊗ eu,nts)
+if i = 0,
+ℏ
+�
+s≥0
+i
+�
+u=1
+(−eu,it−s−1 ⊗ ei+1,uts+1 + ei+1,ut−s ⊗ eu,its)
++ℏ
+�
+s≥0
+n
+�
+u=i+1
+(−eu,it−s ⊗ ei+1,uts + ei+1,ut−s−1 ⊗ eu,its+1)
+if i ̸= 0.
+9
+
+Bi =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏ
+�
+s≥0
+a
+�
+u=b+1
+(eu,nt−s−1en,uts+1 + en,ut−seu,nts)
+−ℏ
+�
+s≥0
+a
+�
+u=b+1
+(eu,1t−s−1e1,uts+1 + e1,ut−seu,1ts)
+−ℏ
+�
+w≤m−n
+W (1)
+w,w + ℏ(a − b)en,nt + ℏ(a − b)ca
+if i = 0,
+ℏ
+�
+s≥0
+a
+�
+u=b+1
+eu,it−s−1ei,uts+1 − ℏ
+�
+s≥0
+a
+�
+u=b+1
+eu,i+1t−sei+1,uts
+if i ̸= 0,
+B+
+i =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏ
+�
+s≥0
+a
+�
+u=b+1
+(eu,1t−s−1en,uts+2 + en,ut1−seu,1ts)
+if i = 0,
+ℏ
+�
+s≥0
+a
+�
+u=b+1
+(eu,i+1t−s−1ei,uts+1 + ei,ut−seu,i+1ts)
+if i ̸= 0,
+B−
+i =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ℏ
+�
+s≥0
+a
+�
+u=n+1
+(eu,nt−s−1e1,uts + e1,ut−1−seu,nts) + ℏ(a − b)e1,nt−1
+if i = 0,
+ℏ
+�
+s≥0
+a
+�
+u=b+1
+(eu,it−s−1ei+1,uts + ei+1,ut−1−seu,its)
+if i ̸= 0.
+The proof of Theorem 3.17 will be written in the appendix. It is enough to show the compatibil-
+ity with (2.2)-(2.11) and (3.6)-(3.15). We only prove that ∆a,b is compatible with [Hi,1, Hj,1] = 0,
+(3.6) and (3.7). We show the compatibility with (3.6) and (3.7) in appendix A and the one with
+[Hi,1, Hj,1] = 0 in appendix B. We can prove the other compatibilities in a similar way.
+Remark 3.18. By the definition of ∆a,b, in the case when a = b = n, we have the following relation;
+(Ψ1 ⊗ Ψ1) ◦ ∆ = ∆n,n ◦ Ψ.
+By this remark, we find that ∆a,b is the natural extension of ∆.
+4
+W-algebras of type A
+We fix some notations for vertex algebras. For a vertex algebra V , we denote the generating field
+associated with v ∈ V by v(z) =
+�
+n∈Z
+v(n)z−n−1. We also denote the OPE of V by
+u(z)v(w) ∼
+�
+s≥0
+(u(s)v)(w)
+(z − w)s+1
+for all u, v ∈ V . We denote the vacuum vector (resp. the translation operator) by |0⟩ (resp. ∂).
+We set
+N =
+l
+�
+i=1
+qi,
+q1 ≥ q2 ≥ · · · ≥ ql.
+10
+
+We set a basis of gl(N) as gl(N) =
+�
+1≤i,j≤N
+Cei,j. We also fix an inner product of gl(N) determined
+by
+(ei,j|ep,q) = kδi,qδp,j + δi,jδp,q.
+col(i) = s if
+s−1
+�
+j=1
+qj < i ≤
+s
+�
+i=1
+qj,
+row(i) = i −
+col(i)−1
+�
+j=1
+qj.
+For all 1 ≤ i, j ≤ N, we take 1 ≤ ˆi,˜i ≤ N as
+col(ˆi) = col(i) + 1, row(ˆi) = row(i),
+col(˜j) = col(j) − 1, row(˜j) = row(j).
+We take a nilpotent element f as
+f =
+�
+1≤j≤N
+eˆj,j.
+We consider two vertex algebras. The first one is the universal affine vertex algebra associated
+with a Lie subalgebra
+b =
+�
+1≤i,j≤N
+col(i)≥col(j)
+Cei,j ⊂ gl(N)
+and its inner product
+κ(ei,j, ep,q) = αcol(i)δi,qδp,j + δi,jδp,q,
+where αi = k + N − qi.
+The second one is the universal affine vertex algebra associated with a Lie superalgebra a =
+b ⊕
+�
+1≤i,j≤N
+col(i)>col(j)
+Cψi,j with the following commutator relations;
+[ei,j, ψp,q] = δj,pψi,q − δi,qψp,j,
+[ψi,j, ψp,q] = 0,
+where ei,j is an even element and ψi,j is an odd element. We set the inner product on a such that
+�κ(ei,j, ep,q) = κ(ei,j, ep,q),
+�κ(ei,j, ψp,q) = �κ(ψi,j, ψp,q) = 0.
+By the definition of V �κ(a) and V κ(b), V �κ(a) contains V κ(b).
+By the PBW theorem, we can identify V �κ(a) (resp. V κ(b)) with U(a[t−1]) (resp. U(b[t−1])).
+In order to simplify the notation, here after, we denote the generating field (ut−1)(z) as u(z). By
+the definition of V �κ(a), generating fields u(z) and v(z) satisfy the OPE
+u(z)v(w) ∼ [u, v](w)
+z − w
++ κ(u, v)
+(z − w)2
+(4.1)
+for all u, v ∈ a.
+For all u ∈ a, let u[−s] be ut−s. In this section, we regard V �κ(a) (resp. V κ(b)) as a non-
+associative superalgebra whose product · is defined by
+u[−w] · v[−s] = (u[−w])(−1)v[−s].
+11
+
+We sometimes omit · and in order to simplify the notation.
+By [16] and [17], a W-algebra
+Wk(gl(N), f) can be realized as a subalgebra of V κ(b).
+Let us define an odd differential d0 : V κ(b) → V �κ(a) determined by
+d01 = 0,
+(4.2)
+[d0, ∂] = 0,
+(4.3)
+[d0, ei,j[−1]] =
+�
+col(i)>col(r)≥col(j)
+er,j[−1]ψi,r[−1] −
+�
+col(j)<col(r)≤col(i)
+ψr,j[−1]ei,r[−1]
++ δ(col(i) > col(j))αcol(i)ψi,j[−2] + ψˆi,j[−1] − ψi,˜j[−1].
+(4.4)
+By using Theorem 2.4 in [15], we can define the W-algebra Wk(g, f) as follows.
+Definition 4.5. The W-algebra Wk(gl(N), f) is the vertex subalgebra of V κ(b) defined by
+Wk(gl(N), f) = {y ∈ V κ(b) ⊂ V �κ(a) | d0(y) = 0}.
+We construct two kinds of elements W (1)
+i,j and W (2)
+i,j .
+Theorem 4.6. Let us set
+W (1)
+p,q =
+�
+1≤i,j≤N,
+row(i)=p,row(j)=q,
+col(i)=col(j)
+ei,j[−1] for ql < p = q ≤ q1 or 1 ≤ p, q ≤ ql,
+W (2)
+p,q =
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+ei,j[−1] −
+�
+col(i)=col(j)
+row(i)=p,row(j)=q
+γcol(i)ei,j[−2]
++
+�
+col(u)=col(j)<col(i)=col(v)
+row(u)=row(v)≤ql
+row(i)=p,row(j)=q
+eu,j[−1]ei,v[−1] −
+�
+col(u)=col(j)≥col(i)=col(v)
+row(u)=row(v)>ql
+row(i)=p,row(j)=q
+eu,j[−1]ei,v[−1]
+for p, q ≤ ql,
+where
+γa =
+l
+�
+u=a+1
+αu.
+Then, the W-algebra Wk(gl(N), f) contains W (1)
+p,q and W (2)
+p,q .
+Proof. It is enough to show that d0(W (r)
+p,q ) = 0. First, we show the case when r = 1. By (4.4), if
+col(i) = col(j), we obtain
+[d0, ei,j[−1]] = ψ�i,j[−1] − ψi,�j[−1].
+(4.7)
+By (4.7), we obtain
+d0(W (1)
+p,q ) =
+�
+1≤i,j≤N,
+row(i)=p,row(j)=q,
+col(i)=col(j)
+(ψ�i,j[−1] − ψi,�j[−1])
+=
+�
+1≤i,j≤N,
+row(i)=p,row(j)=q,
+col(i)=col(j)
+ψ�i,j[−1] −
+�
+1≤i,j≤N,
+row(i)=p,row(j)=q,
+col(i)=col(j)
+ψi,�j[−1].
+(4.8)
+12
+
+In the case when ql < p = q ≥ q1 or p, q ≤ ql, we can rewrite the second term of (4.8) as
+−
+�
+1≤x,y≤N,
+row(x)=p,row(y)=q,
+col(x)=col(y)
+ψ�x,y[−1]
+by setting �x = i, y = �j. Thus, we obtain d0(W (1)
+i,j ) = 0.
+Next, we show the case when r = 2. If col(i) = col(j) + 1 = 2, by (4.4), we also have
+[d0, ei,j[−1]]
+=
+�
+col(r)=col(j)
+er,j[−1]ψi,r[−1] −
+�
+col(r)=col(i)
+ψr,j[−1]ei,r[−1]
++ αcol(i)ψi,j[−2] + ψˆi,j[−1] − ψi,˜j[−1].
+(4.9)
+By the definition of W (2)
+i,j , we can rewrite d0(W (2)
+p,q ) as
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+d0(ei,j[−1]) −
+�
+col(i)=col(j)
+row(i)=p,row(j)=q
+γcol(i)d0(ei,j[−2])
++
+�
+col(u)=col(j)<col(i)=col(v)
+row(u)=row(v)≤ql
+row(i)=p,row(j)=q
+d0(eu,j[−1])ei,v[−1]
++
+�
+col(u)=col(j)<col(i)=col(v)
+row(u)=row(v)≤ql
+row(i)=p,row(j)=q
+eu,j[−1]d0(ei,v[−1])
+−
+�
+col(u)=col(j)≥col(i)=col(v)
+row(u)=row(v)>ql
+row(i)=p,row(j)=q
+d0(eu,j[−1])ei,v[−1]
+−
+�
+col(u)=col(j)≥col(i)=col(v)
+row(u)=row(v)>ql
+row(i)=p,row(j)=q
+eu,j[−1]d0(ei,v[−1]).
+(4.10)
+By (4.9), we obtain
+the first term of (4.10)
+=
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+�
+col(r)=col(j)
+er,j[−1]ψi,r[−1] −
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+�
+col(r)=col(i)
+ψr,j[−1]ei,r[−1]
++
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+αcol(i)ψi,j[−2] +
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+(ψˆi,j[−1] − ψi,˜j[−1]).
+(4.11)
+Similarly to the proof of d0(W (1)
+i,j ) = 0, we find that the last term of the right hand side of (4.11)
+is equal to zero. Then, we have
+the first term of (4.10)
+=
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+�
+col(r)=col(j)
+er,j[−1]ψi,r[−1] −
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+�
+col(r)=col(i)
+ψr,j[−1]ei,r[−1]
++
+�
+col(i)=col(j)+1
+row(i)=p,row(j)=q
+αcol(i)ψi,j[−2].
+(4.12)
+13
+
+By (4.7) and (4.3), we obtain
+the second term of (4.10)
+= −
+�
+col(i)=col(j)
+row(i)=p,row(j)=q
+γcol(i)(ψ�i,j[−2] − ψi,�j[−2])
+=
+�
+col(i)=col(j)
+row(i)=p,row(j)=q
+(γcol(ˆi) − γcol(i))ψ�i,j[−2]
+= −
+�
+col(i)=col(j)
+row(i)=p,row(j)=q
+αcol(ˆi)ψˆi,j[−2].
+(4.13)
+By (4.7), we obtain
+the third term of the right hand side of (4.10)
+=
+�
+col(u)=col(j)<col(i)=col(v)
+row(u)=row(v)≤ql
+row(i)=p,row(j)=q
+(ψˆu,j[−1] − ψu,˜j[−1])ei,v[−1]
+=
+�
+col(u)+1=col(j)+1=col(i)=col(v)
+row(u)=row(v)≤ql
+row(i)=p,row(j)=q
+ψˆu,j[−1]ei,v[−1],
+(4.14)
+the 4-th term of the right hand side of (4.10)
+=
+�
+col(u)=col(j)<col(i)=col(v)
+row(u)=row(v)≤ql
+row(i)=p,row(j)=q
+eu,j[−1](ψˆi,v[−1] − ψi,˜v[−1])
+= −
+�
+col(u)+1=col(j)+1=col(i)=col(v)
+row(u)=row(v)≤ql
+row(i)=p,row(j)=q
+eu,j[−1]ψi,˜v[−1],
+(4.15)
+the 5-th term of the right hand side of (4.10)
+= −
+�
+col(u)=col(j)≥col(i)=col(v)
+ql<row(u)=row(v)≤qcol(j)
+row(i)=p,row(j)=q
+(ψˆu,j[−1] − ψu,˜j[−1])ei,v[−1]
+=
+�
+col(u)=col(j)=col(i)+1=col(v)+1
+ql<row(u)=row(v)≤qcol(j)
+row(i)=p,row(j)=q
+ψu,˜j[−1]ei,v[−1],
+(4.16)
+the 6-th term of the right hand side of (4.10)
+= −
+�
+col(u)=col(j)≥col(i)=col(v)
+ql<row(u)=row(v)≤qcol(j)
+row(i)=p,row(j)=q
+eu,j[−1](ψˆi,v[−1] − ψi,˜v[−1])
+= −
+�
+col(u)=col(j)=col(i)+1=col(v)+1
+ql<row(u)=row(v)≤qcol(j)
+row(i)=p,row(j)=q
+eu,j[−1]ψˆi,v[−1].
+(4.17)
+Here after, in order to simplify the notation, let us denote the i-th term of (the number of the
+equation) by (the number of the equation)i. By a direct computation, we obtain
+(4.11)1 + (4.15) + (4.17) = 0,
+14
+
+(4.11)2 + (4.14) + (4.16) = 0,
+(4.11)3 + (4.13) = 0.
+Then, adding (4.11)-(4.17), we obtain d0(W (2)
+i,j ) = 0.
+Remark 4.18. We have already considered the case when l = 2, q1 = m, q2 = n in [23]. In [23], we
+use the different notations about col(i) and row(i) as follows;
+col(i) =
+�
+1
+if i ≤ m,
+2
+if i > m,
+row(i) =
+�
+i
+if i ≤ m,
+i − n
+if i > m.
+In [23], we give the strong generators of Wk(gl(m + n), f) as follows;
+{W (1)
+i,j | i ≤ m − n, 1 ≤ j ≤ m or i, j > m − n}, {W (2)
+i,j | i > m − n}.
+The elements W (1)
+i,j and W (2)
+i,j in Theorem 4.6 are corresponding to the elements W (1)
+m−i+1,m−j+1
+and W (2)
+m−i+1,m−j+1 in [23].
+5
+The universal enveloping algebra of Wk(gl(N), f)
+Let us recall the definition of a universal enveloping algebra of a vertex algebra in the sense of [10]
+and [20]. For any vertex algebra V , let L(V ) be the Borchards Lie algebra, that is,
+L(V ) = V ⊗C[t, t−1]/Im(∂ ⊗ id + id ⊗ d
+dt),
+(5.1)
+where the commutation relation is given by
+[uta, vtb] =
+�
+r≥0
+�
+a
+r
+�
+(u(r)v)ta+b−r
+for all u, v ∈ V and a, b ∈ Z. Now, we define the universal enveloping algebra of V .
+Definition 5.2 (Section 6 in [20]). We set U(V ) as the quotient algebra of the standard degreewise
+completion of the universal enveloping algebra of L(V ) by the completion of the two-sided ideal
+generated by
+(u(a)v)tb −
+�
+i≥0
+�a
+i
+�
+(−1)i(uta−ivtb+i − (−1)avta+b−iuti),
+(5.3)
+|0⟩t−1 − 1.
+(5.4)
+We call U(V ) the universal enveloping algebra of V .
+In the last of this section, we will consider the universal enveloping algebra of Wk(gl(N), f).
+The projection map from gl(N) to
+l�
+i=1
+gl(qi) induces the injective homomorphism called the Miura
+map (see [16])
+µ: Wk(gl(N), f) → V κ(
+l
+�
+i=1
+gl(qi)).
+Let e(r)
+i,j ts ∈
+�
+1≤i≤l
+U(�gl(qi)) be 1⊗r−1 ⊗ e(r)
+i,j ts ⊗ 1⊗l−r. Let us set the degree of
+�
+1≤i≤l
+U(�gl(qi))
+by
+deg(e(r)
+i,j ts) = s.
+15
+
+Induced by the Miura map µ, we obtain
+�µ: U(Wk(gl(N), f)) → �
+�
+1≤i≤lU(�gl(qi)),
+where �
+�
+1≤i≤lU(�gl(qi)) is the standard degreewise completion of �
+1≤i≤l U(�gl(qi)).
+By the definition of W (1)
+i,j and W (2)
+i,j , we have
+�µ(W (1)
+i,j ts) =
+�
+1≤r≤l
+e(r)
+i,j ts,
+(5.5)
+�µ(W (2)
+i,j ts) =
+n
+�
+r=1
+sγre(r)
+i,j ts−1 +
+�
+s∈Z
+�
+r1<r2
+�
+1≤u≤ql
+e(r1)
+u,j t−se(r2)
+i,u ts
+−
+�
+s∈Z
+�
+r1<r2
+�
+ql<u≤qr2
+e(r1)
+i,u t−se(r2)
+u,j ts
+−
+�
+s≥0
+�
+r≥0
+�
+1≤u≤qr
+(e(r)
+u,jt−s−1e(r)
+i,uts+1 + e(r)
+i,ut−se(r)
+u,jts).
+(5.6)
+Since the Miura map is injective (see [9], [2]), �µ is injective.
+6
+Guay’s affine Yangians and non-rectangular W-algebras
+Similarly to Y a
+ℏ,ε−(a−b)ℏ(�sl(n))�⊗Y b
+ℏ,ε(�sl(n)), we define
+Y qg
+ℏ,ε−(qg−ql)ℏ(�sl(n))�⊗Y qg+1
+ℏ,ε−(qg+1−ql)ℏ(�sl(n))�⊗ · · · �⊗Y ql−1
+ℏ,ε−(ql−1−ql)ℏ(�sl(n))�⊗Y ql
+ℏ,ε(�sl(n)).
+We denote this algebra by �⊗
+l
+i=gY qi
+ℏ,ε−(qi−ql)ℏ(�sl(n)). By Theorem 3.17, ∆qg−1,qg naturally induces
+the homomorphism
+�⊗
+l
+i=g+1Y qi
+ℏ,ε−(qi−ql)ℏ(�sl(n)) → �⊗
+l
+i=gY qi
+ℏ,ε−(qi−ql)ℏ(�sl(n)).
+We denote this homomorphism by ∆qg−1,qg ⊗ id⊗l−g. By the definition of ∆qg−1,qg ⊗ id⊗l−g, we
+have a homomorphism
+∆l : Yℏ,ε(�sl(n)) → �⊗
+l
+i=1Y qi
+ℏ,ε−(qi−ql)ℏ(�sl(n))
+determined by
+∆l = (∆q1,q2 ⊗ idl−2) ◦ (∆q2,q3 ⊗ idl−3) ◦ · · · ◦ (∆ql−2,ql−1 ⊗ id) ◦ ∆ql−1,ql ◦ Ψ1.
+By Theorem (3.16), we have a homomorphism
+�evℏ,ε−(qi−ql)ℏ : Y qi
+ℏ,ε−(qi−ql)ℏ(�sl(n)) → U(�gl(qi))comp
+under the assumption that
+cqi = −ε − (qi − ql)ℏ
+ℏ
+.
+By the definition of �⊗
+l
+i=gY qi
+ℏ,ε−(qi−ql)ℏ(�sl(n)), we find that
+�ev
+−ℏ �l
+v=2 αv
+ℏ,ε−(q1−ql)ℏ ⊗ �ev
+−ℏ �l
+v=3 αv
+ℏ,ε−(q2−ql)ℏ ⊗ · · · ⊗ �ev−ℏαl
+ℏ,ε−(ql−1−ql)ℏ ◦ �ev0
+ℏ,ε.
+induces the homomorphism
+evl : �⊗
+l
+i=1Y qi
+ℏ,ε−(qi−ql)ℏ(�sl(n)) → U(�gl(q1))�⊗ · · · �⊗U(�gl(ql)).
+Here after, we sometimes denote ql by n.
+16
+
+Theorem 6.1. Suppose that n ≥ 3 and − ε
+ℏ = k + N. There exists an algebra homomorphism
+Φ: Yℏ,ε(�sl(n)) → U(gl(N), f))
+determined by
+Φ(Hi,0) =
+
+
+
+
+
+W (1)
+n,n − W (1)
+n,n +
+l
+�
+v=1
+αv
+if i = 0,
+W (1)
+i,i − W (1)
+i+1,i+1i
+if i ̸= 0,
+Φ(X+
+i,0) =
+�
+W (1)
+n,1t
+if i = 0,
+W (1)
+i,i+1
+if i ̸= 0,
+Φ(X−
+i,0) =
+�
+W (1)
+1,nt−1
+if i = 0,
+W (1)
+i+1,i
+if i ̸= 0,
+Φ(Hi,1) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−ℏ(W (2)
+n,nt − W (2)
+1,1 t) − ℏ(
+l−1
+�
+v=1
+αv)W (1)
+n,n
+−ℏ(
+l−1
+�
+v=1
+αv)(
+l
+�
+v=1
+αv) + ℏ(
+l
+�
+v=1
+αv)Φ(H0,0) − ℏW (1)
+n,n(W (1)
+1,1 − (
+l
+�
+v=1
+αv))
+−ℏ
+�
+w≤m−n
+W (1)
+w,w + ℏ
+�
+s≥0
+n
+�
+u=1
+W (1)
+n,ut−sW (1)
+u,nts − ℏ
+�
+s≥0
+n
+�
+u=1
+W (1)
+1,ut−s−1W (1)
+u,1ts+1
+if i = 0,
+−ℏ(W (2)
+i,i t − W (2)
+i+1,i+1t) − i
+2ℏΦ(Hi,0) + ℏW (1)
+i,i W (1)
+i+1,i+1
++ℏ
+�
+s≥0
+i
+�
+u=1
+W (1)
+i,u t−sW (1)
+u,i ts + ℏ
+�
+s≥0
+n
+�
+u=i+1
+W (1)
+i,u t−s−1W (1)
+u,i ts+1
+−ℏ
+�
+s≥0
+i
+�
+u=1
+W (1)
+i+1,ut−sW (1)
+u,i+1ts − ℏ
+�
+s≥0
+n
+�
+u=i+1
+W (1)
+i+1,ut−s−1W (1)
+u,i+1ts+1
+if i ̸= 0,
+Φ(X+
+i,1) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−ℏW (2)
+n,1t2 + ℏ(
+l
+�
+v=1
+αv)Φ(X+
+0,0) + ℏ
+�
+s≥0
+n
+�
+u=1
+W (1)
+n,ut−sW (1)
+u,1ts+1
+if i = 0,
+−ℏW (2)
+i,i+1t − i
+2ℏΦ(X+
+i,0)
++ℏ
+�
+s≥0
+i
+�
+u=1
+W (1)
+i,u t−sW (1)
+u,i+1ts + ℏ
+�
+s≥0
+n
+�
+u=i+1
+W (1)
+i,u t−s−1W (1)
+u,i+1ts+1
+if i ̸= 0,
+Φ(X−
+i,1) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−ℏW (2)
+1,n − ℏ(
+l−1
+�
+v=1
+αv)W (1)
+1,nt−1 − ℏ(
+l
+�
+r=1
+αr)Φ(X−
+0,0) + ℏ
+�
+s≥0
+n
+�
+u=1
+W (1)
+1,ut−s−1W (1)
+u,nts
+if i = 0,
+−ℏW (2)
+i+1,it − i
+2ℏΦ(X−
+i,0)
++ℏ
+�
+s≥0
+i
+�
+u=1
+W (1)
+i+1,ut−sW (1)
+u,i ts + ℏ
+�
+s≥0
+n
+�
+u=i+1
+W (1)
+i+1,ut−s−1W (1)
+u,i ts+1
+if i ̸= 0.
+17
+
+Proof. Since �µ is injective, t is enough to show that
+evl ◦∆l(Ai,r) = �µ ◦ Φ(Ai,r) for all r = 0, 1 and A = H, X±.
+We only show the case when i = 0, r = 1, A = X±. Other cases are proven in a similar way. First,
+we show that
+evl ◦∆l(X+
+0,1) = �µ ◦ Φ(X+
+0,1).
+(6.2)
+By (5.5) and (5.6), the right hand side of (6.2) is equal to
+− ℏ
+�
+w∈Z
+�
+r1<r2
+�
+1≤u≤n
+e(r1)
+u,1 t−w+1e(r2)
+n,u tw + ℏ
+�
+w∈Z
+�
+r1<r2
+�
+n<u≤qr2
+e(r1)
+n,u twe(r2)
+u,1 t−w+1
++ ℏ
+�
+s≥0
+�
+1≤r≤l
+�
+n<u≤qr
+(e(r)
+u,1t−s−1e(r)
+n,uts+2 + e(r)
+n,ut−s+1e(r)
+u,1ts)
+− 2ℏ
+l
+�
+a=1
+(
+l
+�
+r=a+1
+αr)e(a)
+n,1t + ℏ
+l
+�
+a=1
+(
+l
+�
+r=1
+αr)e(a)
+n,1t + ℏ
+�
+s≥0
+l
+�
+r=1
+n
+�
+u=1
+e(r)
+n,ut−se(r)
+u,1ts+1
++ ℏ
+�
+s≥0
+�
+r1<r2
+n
+�
+u=1
+e(r1)
+n,u t−se(r2)
+u,1 ts+1 + ℏ
+�
+s≥0
+�
+r1<r2
+n
+�
+u=1
+e(r1)
+u,1 ts+1e(r2)
+n,u t−s,
+(6.3)
+where the first four terms are derived from −ℏW (1)
+n,1t2. By Theorem 3.16 and 3.17, the left hand
+side of (6.2) is equal to
+ℏ
+l
+�
+r=1
+αre(r)
+n,1t + ℏ
+l
+�
+r=1
+n
+�
+u=1
+e(r)
+n,ut−se(r)
+u,1ts+1 − ℏ
+l
+�
+a=1
+(
+l
+�
+r=a+1
+αr)e(a)
+n,1t
++ ℏ
+l
+�
+a=1
+(
+a−1
+�
+r=1
+αr)e(a)
+n,1t + ℏ
+�
+s≥0
+n
+�
+u=1
+�
+r1<r2
+(−e(r1)
+u,n t−se(r2)
+1,u ts+1 + e(r1)
+1,u t−se(r2)
+u,n ts+1)
++ ℏ
+�
+w∈Z
+�
+r1<r2
+qr2
+�
+u=n+1
+e(r1)
+n,u t−we(r2)
+u,1 tw+1
++ ℏ
+l
+�
+r=1
+�
+s≥0
+a
+�
+u=n+1
+(e(r)
+u,1t−s−1e(r)
+n,uts+2 + e(r)
+n,ut1−se(r)
+u,1ts),
+(6.4)
+where (6.4)1, (6.4)2, (6.4)3 are deduced from the evaluation map, (6.4)4, (6.4)5 are deduced from
+Ai, and other terms are deduced from Bi and Fi. Since the relations
+(6.4)1 + (6.4)3 + (6.4)4 = (6.3)4 + (6.3)5,
+(6.4)2 = (6.3)6,
+(6.4)5 = (6.3)1 + (6.3)6 + (6.3)7,
+(6.4)6 = (6.3)2,
+(6.4)7 = (6.3)3
+hold by a direct computation, we obtain (6.2).
+Next, we show that
+evl ◦∆l(X−
+0,1) = �µ ◦ Φ(X−
+0,1).
+(6.5)
+By (5.5) and (5.6), the right hand side of (6.5) is equal to
+− ℏ
+�
+w∈Z
+�
+r1<r2
+�
+1≤u≤n
+e(r1)
+u,n t−w−1e(r2)
+1,u tw
+18
+
++ ℏ
+�
+w∈Z
+�
+r1<r2
+�
+n<u≤qr2
+e(r1)
+1,u twe(r2)
+u,n t−w−1
++ ℏ
+�
+s≥0
+�
+1≤r≤l
+�
+n<u≤qr
+(e(r)
+u,nt���s−1e(r)
+1,uts + e(r)
+1,ut−s−1e(r)
+u,nts) − ℏ
+l
+�
+a=1
+(
+l−1
+�
+r=1
+αr)e(a)
+1,nt−1
++ ℏ
+l
+�
+a=1
+(
+l
+�
+r=1
+αr)e(a)
+1,nt−1 + ℏ
+�
+s≥0
+l
+�
+r=1
+n
+�
+u=1
+e(r)
+1,ut−s−1e(r)
+u,nts
++ ℏ
+�
+s≥0
+�
+r1<r2
+n
+�
+u=1
+e(r1)
+1,u t−s−1e(r2)
+u,n ts + ℏ
+�
+s≥0
+�
+r1<r2
+n
+�
+u=1
+e(r1)
+u,n tse(r2)
+1,u t−s−1,
+(6.6)
+where the first three terms are derived from −ℏW (2)
+1,n. By Theorem 3.16 and 3.17, the left hand
+side of (6.5) is equal to
+ℏ
+l
+�
+r=1
+αre(r)
+1,nt−1 + ℏ
+l
+�
+r=1
+n
+�
+u=1
+e(r)
+1,ut−s−1e(r)
+u,nts − ℏ
+l
+�
+a=1
+(
+l
+�
+r=a+1
+αr)e(a)
+1,nt−1
++ ℏ
+l
+�
+a=1
+(
+l
+�
+r=a+1
+αr)e(a)
+1,nt−1 + ℏ
+�
+s≥0
+n
+�
+u=1
+�
+r1<r2
+(−e(r1)
+u,n t−s−1e(r2)
+1,u ts + e(r1)
+1,u t−s−1e(r2)
+u,n ts)
++ ℏ
+�
+s≥0
+�
+r1<r2
+qr2
+�
+u=n+1
+e(r1)
+1,u t−w−1e(r2)
+u,n tw
++ ℏ
+l
+�
+r=1
+�
+s≥0
+a
+�
+u=n+1
+(e(r)
+u,nt−s−1e(r)
+1,uts + e(r)
+1,ut−s−1eu,nts)
+− ℏ
+l
+�
+a=1
+(αa − αl)e(a)
+1,nt−1,
+(6.7)
+where (6.6)1, (6.6)2, (6.6)3 are deduced from the evaluation map, (6.6)4, (6.6)5 are deduced from
+Ai, and other terms are deduced from Bi and Fi. Since the relations
+(6.7)1 + (6.7)3 + (6.7)4 + (6.7)8 = (6.6)4 + (6.6)5,
+(6.7)2 = (6.6)6,
+(6.7)5 = (6.6)1 + (6.6)8 + (6.6)7,
+(6.7)6 = (6.6)2,
+(6.7)7 = (6.6)3
+hold by a direct computation, we obtain (6.5).
+Remark 6.8. In Theorem 5.1 in [23], we have constructed an algebra homomorphism
+�Φ: Yℏ,ε(�sl(n)) → U(Wk(gl(m + n), f))
+in the case when q1 = m, q2 = n. By the definition of Φ and �Φ, we find that �Φ is different from Φ.
+However, in the same computation to the one of the proof of Theorem 5.1 in [23], we can prove
+that Φ is compatible with the defining relations (2.2)-(2.11).
+A
+The proof of the compatibility with (3.6) and (3.7)
+Take n < u ≤ b. By the definition of ∆a,b, we have
+[∆a,b(Hi,1), ∆a,b(eu,jtx)]
+19
+
+= [Hi,1, eu,jtx] ⊗ 1 + 1 ⊗ [Hi,1, eu,jtx] − [Fi, □(eu,jtx)] + [Ai, □(eu,jtx)] + [Bi, □(eu,jtx)],
+where □(y) = y ⊗ 1 + 1 ⊗ y. Thus, it is enough to show that
+(∆a,b − □)([Hi,1eu,jtx]) = −[Fi, □(eu,jtx)] + [Ai, □(eu,jtx)] + [Bi, □(eu,jtx)].
+First, let us compute [Fi, □(eu,jtx)]. By a direct computation, we obtain
+[ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,itw ⊗ ei,vt−w, □(eu,jtx)]
+= ℏ
+�
+w∈Z
+eu,itw ⊗ ei,jtx−w − δi,jℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,itw ⊗ eu,vtx−w
+− α2δi,jℏxeu,itx ⊗ 1.
+(A.1)
+Here after, we denote by (equation number)p,q the formula that substitutes i = p, j = q for the
+right hand side of (equation number). By the definition of Fi, we find that
+[Fi, □(eu,jtx)] = (A.1)i,j − (A.1)i+1,j.
+Next, let us compute [Ai, □(eu,jtx)]. By a direct computation, we obtain
+[−ℏ(ei,i ⊗ ei+1,i+1 + ei+1,i+1 ⊗ ei,i), □(eu,jtx)]
+= ℏδj,i+1ei,i ⊗ eu,jtx + ℏδi,jeu,jtx ⊗ ei+1,i+1
++ ℏδi,jei+1,i+1 ⊗ eu,jtx + ℏδj,i+1eu,jtx ⊗ ei,i,
+(A.2)
+[ℏ
+�
+s≥0
+i
+�
+k=1
+(−ek,it−s−1 ⊗ ei,kts+1 + ei,kt−s ⊗ ek,its), □(eu,jtx)]
+= ℏ
+�
+s≥0
+δ(j ≤ i)eu,ita−s−1 ⊗ ei,jts+1 + δi,jℏ
+�
+s≥0
+i
+�
+k=1
+ek,it−s−1 ⊗ eu,kts+x+1
+− δi,jℏ
+�
+s≥0
+i
+�
+k=1
+eu,ktx−s ⊗ ek,its − ℏ
+�
+s≥0
+δ(j ≤ i)ei,jt−s ⊗ eu,its+x,
+(A.3)
+[ℏ
+�
+s≥0
+n
+�
+k=i+1
+(−ek,it−s ⊗ ei,kts + ei,kt−s−1 ⊗ ek,its+1), □(eu,jtx)]
+= ℏ
+�
+s≥0
+δ(j > i)eu,itx−s ⊗ ei,jts + δi,jℏ
+�
+s≥0
+n
+�
+k=i+1
+ek,it−s ⊗ eu,kts+x
+− δi,jℏ
+�
+s≥0
+n
+�
+k=i+1
+eu,ktx−s−1 ⊗ ek,its+1 − ℏ
+�
+s≥0
+δ(j > i)ei,jt−s−1 ⊗ eu,its+x+1,
+(A.4)
+− [ℏ
+�
+s≥0
+i
+�
+k=1
+(−ek,i+1t−s−1 ⊗ ei+1,kts+1 + ei+1,kt−s ⊗ ek,its), □(eu,jtx)]
+= −ℏ
+�
+s≥0
+δ(j ≤ i)eu,i+1tx−s−1 ⊗ ei+1,jts+1 − δi+1,jℏ
+�
+s≥0
+i
+�
+k=1
+ek,i+1t−s−1 ⊗ eu,kts+x+1
+20
+
++ δi+1,jℏ
+�
+s≥0
+i
+�
+k=1
+eu,ktx−s ⊗ ek,its + ℏ
+�
+s≥0
+δ(j ≤ i)ei+1,jt−s ⊗ eu,i+1ts+x,
+(A.5)
+− [ℏ
+�
+s≥0
+n
+�
+k=i+1
+(−ek,it−s ⊗ ei,kts + ei,kt−s−1 ⊗ ek,its+1), □(eu,jtx)]
+= −ℏ
+�
+s≥0
+δ(j > i)eu,i+1tx−s ⊗ ei+1,jts − δi,jℏ
+�
+s≥0
+n
+�
+k=i+1
+ek,i+1t−s ⊗ eu,kts+x
++ δi,jℏ
+�
+s≥0
+n
+�
+k=i+1
+eu,ktx−s−1 ⊗ ek,i+1ts+1 + ℏ
+�
+s≥0
+δ(j > i)ei+1,jt−s−1 ⊗ eu,i+1ts+x+1.
+(A.6)
+By the definition of Ai, we find that
+[Ai, □(eu,jtx)] = (A.2)i,j + (A.3)i,j + (A.4)i,j + (A.5)i,j + (A.6)i,j.
+Finally, let us compute [Bi ⊗ 1, □(eu,jta)]. By a direct computation, we obtain
+[ℏ
+�
+s≥0
+a
+�
+v=b+1
+(ev,it−s−1ei,vts+1 + ei,vt−sev,its), eu,jtx]
+= −ℏδi,j
+�
+s≥0
+a
+�
+v=b+1
+(ev,it−s−1eu,vtx+s+1 + eu,vtx−sev,its).
+(A.7)
+By the definition of Bi, we have
+[Bi ⊗ 1, □(eu,jtx)] = (A.7)i,j − (A.7)i+1,j
+Here after, we denote by (equation number)p,q,m m-th term of the right hand side of the
+formula that substitutes i = p, j = q for the right hand side of (equation number).
+Since we obtain
+− (A.1)i,j,1 + (A.3)i,j,1 + (A.4)i,j,1
+= −ℏ
+�
+s≥0
+δ(j ≤ i)eu,its+x ⊗ ei,jt−s − ℏ
+�
+s≥0
+δ(j > i)eu,itx+s+1 ⊗ ei,jt−s−1.
+by a direct computation, we have
+− (A.1)i,j,1 + (A.3)i,j,1 + (A.4)i,j,1 + (A.3)i,j,4 + (A.4)i,j,4
+= −(∆a,b − □)(ℏ
+�
+s≥0
+δ(j ≤ i)ei,jt−seu,its+x) − (∆a,b − □)(ℏ
+�
+s≥0
+δ(j > i)ei,kt−s−1eu,itx+s+1).
+(A.8)
+Since we also obtain
+(A.1)i+1,j,1 + (A.5)i,j,1 + (A.6)i,j,1
+= ℏ
+�
+s≥0
+δ(j ≤ i)eu,i+1ts+x ⊗ ei+1,jt−s + ℏ
+�
+s≥0
+δ(j > i)eu,i+1tx+s+1 ⊗ ei+1,jt−s−1.
+by a direct computation, we find that
+(A.1)i+1,j,1 + (A.5)i,j,1 + (A.6)i,j,1 + (A.5)i,j,4 + (A.6)i,j,4
+= (∆a,b − □)(ℏ
+�
+s≥0
+δ(j ≤ i)ei+1,jt−seu,i+1ts+x)
+21
+
++ (∆a,b − □)(ℏ
+�
+s≥0
+δ(j > i)ei+1,jt−s−1eu,i+1tx+s+1).
+(A.9)
+First, let us show the compatibility with (3.6). In the case when i ̸= j, j + 1, we obtain
+[∆a,b(Hi,1), ∆a,b(eu,jta)]
+= −(A.1)i,j,1 + (A.3)i,j,1 + (A.4)i,j,1 + (A.3)i,j,4 + (A.4)i,j,4
++ (A.1)i+1,j,1 + (A.5)i,j,1 + (A.6)i,j,1 + (A.5)i,j,4 + (A.6)i,j,4.
+Thus, by (A.8) and (A.9), we find that the compatibility with (3.6).
+Next, we show the compatibility with (3.7). We write down (3.7) as follows;
+[Hi−1,1, eu,itx] + [Hi,1, eu,itx]
+= ℏ
+2eu,itx + ℏei−1,i−1eu,itx + ℏeu,itxei+1,i+1
+− ℏ
+�
+s≥0
+ei−1,it−s−1eu,i−1ts+x+1 + ℏ
+�
+s≥0
+ei+1,it−seu,i+1ts+w
+− ℏeu,itxei,i − ℏei,ieu,itx.
+By the definition of ∆a,b, we obtain
+[∆a,b(Hi−1,1), ∆a,b(eu,itx)] + [∆a,b(Hi,1), ∆a,b(eu,itx)]
+= (A.1)i−1,i − (A.1)i,i + (A.1)i,i − (A.1)i+1,i
++ (A.2)i−1,i + (A.3)i−1,i + (A.4)i−1,i + (A.5)i−1,i + (A.6)i−1,i
++ (A.2)i,i + (A.3)i,i + (A.4)i,i + (A.5)i,i + (A.6)i,i
++ (A.7)i−1,i − (A.7)i,i + (A.7)i,i − (A.7)i+1,i.
+By a direct computation, we obtain
+(A.1)i−1,i,2 − (A.1)i,i,2 = 0,
+(A.1)i−1,i,3 − (A.1)i,i,3 = 0,
+(A.7)i−1,i − (A.7)i,i + (A.7)i,i − (A.7)i+1,i = 0.
+By a direct computation, we obtain
+(A.2)i−1,i,1 + (A.2)i,i,2 = ℏ(∆a,b − □)ei−1,i−1eu,itx + ℏ(∆a,b − □)eu,itxei+1,i+1.
+(A.10)
+By using
+(A.5)i−1,i,2 − (A.3)i,i,2 = ℏ
+�
+s≥0
+ei,it−s−1 ⊗ eu,its+x+1,
+(A.6)i−1,i,2 − (A.4)i,i,2 = −ℏ
+�
+s≥0
+ei,it−s ⊗ eu,its+x,
+we obtain
+(A.5)i−1,i,2 − (A.3)i,i,2 + (A.4)i−1,i,2 − (A.6)i,i,2 = −ℏei,i ⊗ eu,itx.
+(A.11)
+By using
+(A.5)i−1,i,3 − (A.3)i,i,3 = −ℏ
+�
+s≥0
+eu,itx−s ⊗ ei,its,
+(A.6)i−1,i,3 − (A.4)i,i,3 = ℏ
+�
+s≥0
+eu,itx−s−1 ⊗ ei,its+1,
+22
+
+we obtain
+(A.5)i−1,i,3 − (A.3)i,i,3 + (A.6)i−1,i,3 − (A.4)i,i,3 = −ℏeu,itx ⊗ ei,i.
+(A.12)
+By (A.11) and (A.12), we have
+(A.3)i−1,i,2 − (A.3)i,i,2 + (A.4)i−1,i,2 − (A.4)i,i,2
++ (A.3)i−1,i,3 − (A.3)i,i,3 + (A.4)i−1,i,3 − (A.4)i,i,3
+= −(∆a,b − □)(ℏeu,itxei,i).
+(A.13)
+By a direct computation, we obtain
+(A.8)i,i + (A.9)i−1,i = −(∆a,b − □)(ℏei,ieu,itx).
+(A.14)
+By (A.8), (A.9), (A.10), (A.12), (A.13) and (A.14), we find the compatibility with (3.7).
+B
+The proof of compatibility with [Hi,1, Hj,1] = 0
+By the definition of Ai, Fi and Bi, we find that
+[∆a,b(Hi,1), ∆a,b(Hj,1)]
+= [Hi,1 + Bi, Hj,1 + Bj] ⊗ 1 + [(Hi,1 + Bi) ⊗ 1, Aj] − [(Hj,1 + Bj) ⊗ 1, Ai]
++ [1 ⊗ Hi,1, Aj] − [1 ⊗ Hj,1, Ai] + [Ai, Aj]
+− [(Hi,1 + Bi) ⊗ 1, Fj] + [(Hj,1 + Bj) ⊗ 1, Fi]
+− [1 ⊗ Hi,1, Fj] + [1 ⊗ Hj,1, Fi] − [Ai, Fj] + [Aj, Fi] + [Fi, Fj].
+(B.1)
+By a direct computation, we obtain
+[Fi, Fj] = 0.
+(B.2)
+By a similar proof of Theorem 5.2 in [13], we have
+[Hi,1 ⊗ 1, Aj] − [Hj,1 ⊗ 1, Ai] + [1 ⊗ Hi,1, Aj] − [1 ⊗ Hj,1, Ai] + [Ai, Aj] = 0.
+(B.3)
+By (B.1), (B.2) and (B.3), it is enough to show the following two lemmas.
+Lemma B.4. The following equation holds;
+[Hi,1 + Bi, Hj,1 + Bj] ⊗ 1 + [Bi ⊗ 1, Aj] − [Bj ⊗ 1, Ai].
+(B.5)
+Lemma B.6. The following equation holds;
+[(Hi,1+Bi)⊗1, Fj]−[(Hj,1+Bj)⊗1, Fi]+[1⊗Hi,1, Fj]−[1⊗Hj,1, Fi]+[Ai, Fj]−[Aj, Fi] = 0. (B.7)
+B.1
+The proof of Lemma B.4
+In this subsection, we prove Lemma B.4. Let us consider the first term of the left hand side of
+(B.3). By the defining relation [Hi,1, Hj,1] = 0, we obtain
+the first term of the left hand side of (B.3)
+= [Hi,1, Bj] − [Hj,1, Bi] + [Bi, Bj].
+By the defining relations (3.6)-(3.15) and the form of Bi, it is no problem to assume that
+[Hi,1, ev,jtw] = [evℏ,ε−(a−b)ℏ(Hi,1), ev,jtw],
+[Hi,1, ej,vtw] = [evℏ,ε−(a−b)ℏ(Hi,1), ej,vtw]
+23
+
+hold in Y a
+ℏ,ε−(a−b)ℏ(�sl(n)). Thus, it is enough to show that
+[evℏ,ε−(a−b)ℏ(Hi,1) + Bi, evℏ,ε−(a−b)ℏ(Hj,1) + Bj] ⊗ 1 + [Bi ⊗ 1, Aj] − [Bj ⊗ 1, Ai]
+(B.8)
+is equal to zero. By Theorem 5.1 in [23] and Remark 6.8, (B.8) holds in the case when b = n.
+Comparing the two cases when b = n and b > n, the difference of (B.8) comes from the difference
+of the inner form. In the computation of (B.8), the terms affected by the inner product are
+�
+s1,s2≥0
+a
+�
+u1=b+1
+a
+�
+u2=b+1
+eu1,it−s1−1(ei,u1ts1+1, eu2,jt−s1−1)ej,u2ts2+1
++
+�
+s1,s2≥0
+a
+�
+u1=b+1
+a
+�
+u2=b+1
+eu2,jt−s2−1(eu1,it−s1−1, ej,u2ts2+1)ei,u1ts1+1
+and
+�
+s1,s2≥0
+a
+�
+u1=b+1
+a
+�
+u2=b+1
+ei,u1t−s1(eu1,its1, ej,u2t−s2)eu2,jts2
++
+�
+s1,s2≥0
+a
+�
+u1=b+1
+a
+�
+u2=b+1
+ej,u2t−s2(ei,u1t−s1, eu2,jts2)eu1,its1,
+where ( , ) is an inner product on U(�gl(a)ca). By a direct computation, these terms are equal to
+zero. Thus, we find that (B.8) is equal to zero.
+B.2
+The proof of Lemma B.6
+In this subsection, we prove Lemma B.6. By the similar discussion to the one in the previous
+subsection, it is no problem to assume that
+[Hi,1, ev,jtw] = [evℏ,ε(Hi,1), ev,jtw],
+[Hi,1, ej,vtw] = [evℏ,ε(Hi,1), ej,vtw].
+hold in Y a
+ℏ,ε(�sl(n)). We only prove the case when i < j. Let us compute [Ai, Fj]. By a direct
+computation, we obtain
+− [ℏ(ei+1,i+1 ⊗ ei,i + ei,i ⊗ ei+1,i+1), ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,jtw ⊗ ej,vt−w]
+= ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δj,i+1ev,jtw ⊗ ei,iej,vt−w − ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δi,jev,jtwei+1,i+1 ⊗ ej,vt−w
++ ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δj,iev,jtw ⊗ ei+1,i+1ej,vt−w − ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δj,i+1ev,jtwei,i ⊗ ej,vt−w,
+(B.9)
+[ℏ
+�
+s≥0
+i
+�
+u=1
+(−eu,it−s−1 ⊗ ei,uts+1 + ei,ut−s ⊗ eu,its), ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,jtw ⊗ ej,vt−w]
+= −ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j ≤ i)ej,it−s−1ev,jtw ⊗ ei,vts−w+1
++ ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j ≤ i)ev,it−s+w−1 ⊗ ej,vt−wei,jts+1
+24
+
+− ℏ2 �
+s≥0
+�
+w∈Z
+i
+�
+u=1
+b
+�
+v=n+1
+δi,jev,ut−s+w ⊗ eu,itsej,vt−w
++ ℏ2 �
+s≥0
+�
+w∈Z
+i
+�
+u=1
+b
+�
+v=n+1
+δi,jev,jtwei,ut−s ⊗ eu,vts−w,
+(B.10)
+[ℏ
+�
+s≥0
+n
+�
+u=i+1
+(−eu,it−s ⊗ ei,uts + ei,ut−s−1 ⊗ eu,its+1), ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,jtw ⊗ ej,vt−w]
+= −ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j > i)ej,it−sev,jtw ⊗ ei,vts−w
++ ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j > i)ev,it−s+w ⊗ ej,vt−wei,jts
+− ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=i+1
+b
+�
+v=n+1
+δi,jev,ut−s+w−1 ⊗ eu,its+1ej,vt−w
++ ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=i+1
+b
+�
+v=n+1
+δi,jev,jtwej,ut−s−1 ⊗ eu,vts−w+1,
+(B.11)
+− [ℏ
+�
+s≥0
+i
+�
+u=1
+(−eu,i+1t−s−1 ⊗ ei+1,uts+1 + ei+1,ut−s ⊗ eu,i+1ts), ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,jtw ⊗ ej,vt−w]
+= ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j ≤ i)ej,i+1t−s−1ev,jtw ⊗ ei+1,vts−w+1
+− ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j ≤ i)ev,i+1t−s+w−1 ⊗ ej,vt−wei+1,jts+1
++ ℏ2 �
+s≥0
+�
+w∈Z
+i
+�
+u=1
+b
+�
+v=n+1
+δi+1,jev,ut−s+w ⊗ eu,i+1tsej,vt−w
+− ℏ2 �
+s≥0
+�
+w∈Z
+i
+�
+u=1
+b
+�
+v=n+1
+δi+1,jev,jtwei+1,ut−s ⊗ eu,vts−w,
+(B.12)
+− [ℏ
+�
+s≥0
+n
+�
+u=i+1
+(−eu,i+1t−s ⊗ ei+1,uts + ei+1,ut−s−1 ⊗ eu,i+1ts+1, ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,jtw ⊗ ej,vt−w]
+= ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j > i)ej,i+1t−sev,jtw ⊗ ei+1,vts−w
+− ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j > i)ev,i+1t−s+w ⊗ ej,vt−wei+1,jts
++ ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=i+1
+b
+�
+v=n+1
+δi+1,jev,ut−s+w−1 ⊗ eu,i+1ts+1ej,vt−w
+− ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=i+1
+b
+�
+v=n+1
+δi+1,jev,jtwei+1,ut−s−1 ⊗ eu,vts−w+1.
+(B.13)
+25
+
+By the definition of Ai and Fi, we obtain
+[Ai, Fj] − [Aj, Fi]
+= (B.9)i,j − (B.9)i,j+1 − (B.9)j,i + (B.9)j,i+1
++ (B.10)i,j − (B.10)i,j+1 − (B.10)j,i + (B.10)j,i+1
++ (B.11)i,j − (B.11)i,j+1 − (B.11)j,i + (B.11)j,i+1
++ (B.12)i,j − (B.12)i,j+1 − (B.12)j,i + (B.12)j,i+1
++ (B.13)i,j − (B.13)i,j+1 − (B.13)j,i + (B.13)j,i+1.
+By the assumption i < j, we obtain
+[Ai, Fj] − [Aj, Fi]
+= (B.9)i,j,1 + (B.9)j,i+1,2 + (B.9)j,i+1,3 + (B.9)i,j,4
+− (B.10)j,i,1 + (B.10)j,i+1,1 − (B.10)j,i,2 + (B.10)j,i+1,2 + (B.10)j,i+1,3 + (B.10)j,i+1,4
++ (B.11)i,j,1 − (B.11)i,j+1,1 + (B.11)i,j,2 − (B.11)i,j+1,2 + (B.11)j,i+1,3 + (B.11)j,i+1,4
+− (B.12)j,i,1 + (B.12)j,i+1,1 − (B.12)j,i,2 + (B.12)j,i+1,2 + (B.12)i,j,3 + (B.12)i,j,4
++ (B.13)i,j,1 − (B.13)i,j+1,1 + (B.13)i,j,2 − (B.13)i,j+1,2 + (B.13)i,j,3 + (B.13)i,j,4.
+By a direct computation, we obtain
+(B.12)i,j,4 + (B.10)j,i+1,4
+= −ℏ2 �
+s≥0
+�
+w∈Z
+i
+�
+u=1
+b
+�
+v=n+1
+δi+1,jev,i+1twei+1,ut−s ⊗ eu,vts−w
++ ℏ2 �
+s≥0
+�
+w∈Z
+j
+�
+u=1
+b
+�
+v=n+1
+δj,i+1ev,jtwej,ut−s ⊗ eu,vts−w
+= ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δj,i+1ev,jtwej,jt−s ⊗ ej,vts−w,
+(B.14)
+(B.13)i,j,4 + (B.11)j,i+1,4
+= −ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=i+1
+b
+�
+v=n+1
+δi+1,jev,i+1twei+1,ut−s−1 ⊗ eu,vts−w+1
++ ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=j+1
+b
+�
+v=n+1
+δj,i+1ev,jtwej,ut−s−1 ⊗ eu,vts−w+1
+= −ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,i+1t−s−1ei+1,i+1tw ⊗ ei+1,vts−w+1.
+(B.15)
+By adding (B.14) and (B.15), we have
+(B.12)i,j,4 + (B.10)j,i+1,4 + (B.13)i,j,4 + (B.11)j,i+1,4
+= ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δj,i+1ev,jtwej,j ⊗ ej,vt−w.
+(B.16)
+Similarly to (B.16), we obtain
+(B.12)i,j,3 + (B.10)j,i+1,3
+26
+
+= ℏ2 �
+s≥0
+�
+w∈Z
+i
+�
+u=1
+a
+�
+v=n+1
+δi+1,jev,ut−s+w ⊗ eu,i+1tsei+1,vt−w
+− ℏ2 �
+s≥0
+�
+w∈Z
+j
+�
+u=1
+a
+�
+v=n+1
+δj,i+1ev,ut−s+w ⊗ eu,jtsej,vt−w
+= −ℏ2 �
+s≥0
+�
+w∈Z
+a
+�
+v=n+1
+δj,i+1ev,jt−s+w ⊗ ej,jtsej,vt−w,
+(B.17)
+(B.13)i,j,3 + (B.11)j,i+1,3
+= ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=i+1
+a
+�
+v=n+1
+δi+1,jev,ut−s+w−1 ⊗ eu,i+1ts+1ei+1,vt−w
+− ℏ2 �
+s≥0
+�
+w∈Z
+n
+�
+u=j+1
+a
+�
+v=n+1
+δj,i+1ev,ut−s+w−1 ⊗ eu,jts+1ej,vt−w
+= ℏ2 �
+s≥0
+�
+w∈Z
+a
+�
+v=n+1
+δi+1,jev,i+1t−s+w−1 ⊗ ei+1,i+1ts+1ei+1,vt−w.
+(B.18)
+By adding (B.17) and (B.18), we have
+(B.12)i,j,3 + (B.10)j,i+1,3 + (B.13)i,j,3 + (B.11)j,i+1,3
+= −ℏ2 �
+w∈Z
+a
+�
+v=n+1
+δj,i+1ev,jtw ⊗ ej,jej,vt−w.
+(B.19)
+Then, we obtain
+[Ai, Fj] − [Aj, Fi]
+= (B.9)i,j,1 + (B.9)i,j,4 + (B.9)j,i+1,2 + (B.9)j,i+1,3
+− (B.10)j,i,1 − (B.10)j,i,2 + (B.10)j,i+1,1 + (B.10)j,i+1,2
++ (B.11)i,j,1 + (B.11)i,j,2 − (B.11)i,j+1,1 − (B.11)i,j+1,2
+− (B.12)j,i,1 − (B.12)j,i,2 + (B.12)j,i+1,1 + (B.12)j,i+1,2
++ (B.13)i,j,1 + (B.13)i,j,2 − (B.13)i,j+1,1 − (B.13)i,j+1,2 + (B.16) + (B.19).
+Next, let us compute [(Hi,1 + Bi) ⊗ 1, Fj] − [(Hj,1 + Bj) ⊗ 1, Fi]. By the defining relation, By
+a direct computation, we obtain
+[(Hi,1 + Bi) ⊗ 1, ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,jtw ⊗ ej,vt−w]
+= i
+2
+�
+w∈Z
+b
+�
+v=n+1
+ℏ2δi,jev,jtw ⊗ ej,vt−w − i
+2
+�
+w∈Z
+b
+�
+v=n+1
+ℏ2δi+1,jev,jtw ⊗ ej,vt−w
++
+�
+w∈Z
+b
+�
+v=n+1
+ℏ2δi,jev,jtwei+1,i+1 ⊗ ej,vt−w
++
+�
+w∈Z
+b
+�
+v=n+1
+ℏ2δi+1,jei,iev,jtw ⊗ ej,vt−w
+− ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δ(j ≤ i)
+�
+s≥0
+ei,jt−sev,its+w ⊗ ej,vt−w
+27
+
+− ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+i
+�
+u=1
+δi,jev,utw−seu,its ⊗ ej,vt−w
+− ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+δ(j > i)ei,jt−s−1ev,its+w+1 ⊗ ej,vt−w
+− ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+n
+�
+u=i+1
+δi,jev,utw−s−1eu,its+1 ⊗ ej,vt−w
++ ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+i
+�
+u=1
+δ(j ≤ i)ei+1,jt−sev,i+1ts+w ⊗ ej,vt−w
++ ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+i
+�
+u=1
+δi+1,jev,utw−seu,i+1ts ⊗ ej,vt−w
++ ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+δ(j > i)ei+1,jt−s−1ev,i+1ts+w+1 ⊗ ej,vt−w
++ ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+n
+�
+u=i+1
+δi+1,jev,utw−s−1eu,i+1ts+1 ⊗ ej,vt−w
+− δi,jℏ2 �
+s≥0
+a
+�
+u=b+1
+�
+w∈Z
+b
+�
+v=n+1
+eu,it−s−1ev,uts+w+1 ⊗ ej,vt−w
+− δi,jℏ2 �
+s≥0
+a
+�
+u=b+1
+�
+w∈Z
+b
+�
+v=n+1
+ev,utw−seu,its ⊗ ej,vt−w
++ δi+1,jℏ2 �
+s≥0
+a
+�
+u=b+1
+�
+w∈Z
+b
+�
+v=n+1
+eu,i+1t−s−1ev,uts+w+1 ⊗ ej,vt−w
+− δi+1,jℏ2 �
+s≥0
+a
+�
+u=b+1
+�
+w∈Z
+b
+�
+v=n+1
+ev,utw−seu,i+1ts ⊗ ej,vt−w.
+(B.20)
+By the assumption i < j, we have
+[(Hi,1 + Bi) ⊗ 1, Fj] − [(Hj,1 + Bi) ⊗ 1, Fi]
+= (B.20)j,i+1,1 + (B.20)i,j,2 + (B.20)j,i+1,3 + (B.20)j,i+1,4
+− (B.20)j,i,5 + (B.20)j,i+1,5 + (B.20)j,i+1,6
++ (B.20)i,j,7 − (B.20)i,j+1,7 + (B.20)j,i+1,8 − (B.20)j,i,9 + (B.20)j,i+1,9 + (B.20)i,j,10
++ (B.20)i,j,11 − (B.20)i,j+1,11 + (B.20)i,j,12
++ (B.20)j,i+1,13 + (B.20)j,i+1,14 + (B.20)i,j,15 + (B.20)i,j,16.
+By a direct computation, we obtain
+(B.20)j,i+1,1 + (B.20)i,j,2
+= j
+2
+�
+w∈Z
+b
+�
+v=n+1
+ℏ2δj,i+1ev,i+1tw ⊗ ei+1,vt−w − i
+2
+�
+w∈Z
+b
+�
+v=n+1
+ℏ2δi+1,jev,jtw ⊗ ej,vt−w
+= 1
+2
+�
+w∈Z
+b
+�
+v=n+1
+ℏ2δj,i+1ev,i+1tw ⊗ ei+1,vt−w,
+(B.21)
+28
+
+By a direct computation, we obtain
+(B.20)j,i+1,13 + (B.20)i,j,15 = 0,
+(B.22)
+(B.20)j,i+1,14 + (B.20)i,j,16 = 0.
+(B.23)
+By using
+(B.20)j,i+1,6 + (B.20)i,j,10
+= −ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+j
+�
+u=1
+δj,i+1ev,utw−seu,jts ⊗ ei+1,vt−w
++ ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+i
+�
+u=1
+δi+1,jev,utw−seu,i+1ts ⊗ ej,vt−w
+= −ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+δj,i+1ev,jtw−sej,jts ⊗ ei+1,vt−w
+and
+(B.20)j,i+1,8 + (B.20)i,j,12
+= −ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+n
+�
+u=j+1
+δi+1,jei+1,vt−w ⊗ ev,utw−s−1eu,jts+1
++ ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+n
+�
+u=i+1
+δi+1,jev,utw−s−1eu,i+1ts+1 ⊗ ej,vt−w
+= ℏ2 �
+w∈Z
+b
+�
+v=n+1
+�
+s≥0
+δi+1,jev,i+1tw−s−1ei+1,i+1ts+1 ⊗ ej,vt−w,
+we find that
+(B.20)j,i+1,6 + (B.20)i,j,10 + (B.20)j,i+1,8 + (B.20)i,j,12
+= −ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δj,i+1ev,jtwej,j ⊗ ei+1,vt−w.
+(B.24)
+By a direct computation, we obtain
+(B.16) + (B.24) = 0,
+(B.25)
+(B.9)j,i+1,2 + (B.20)j,i+1,3 = 0,
+(B.26)
+(B.9)i,j,4 + (B.20)i,j,4 = 0,
+(B.27)
+− (B.10)j,i,1 + (B.20)i,j,7 = 0,
+(B.28)
+(B.11)i,j,1 − (B.20)j,i,5 = 0,
+(B.29)
+− (B.12)j,i,1 − (B.20)i,j+1,7 = 0,
+(B.30)
+(B.13)i,j,1 + (B.20)j,i+1,5 = 0,
+(B.31)
+(B.10)j,i+1,1 + (B.20)i,j,11 = 0,
+(B.32)
+− (B.11)i,j+1,1 − (B.20)j,i,9 = 0,
+(B.33)
+(B.12)j,i+1,1 − (B.20)i,j+1,11 = 0,
+(B.34)
+− (B.13)i,j+1,1 + (B.20)j,i+1,9 = 0.
+(B.35)
+29
+
+Then, we find that
+[(Hi,1 + Bi) ⊗ 1, Fj] − [(Hj,1 + Bj) ⊗ 1, Fi]
+= (B.9)i,j,1 + (B.9)i,j,4 − (B.10)j,i,2 + (B.10)j,i+1,2
++ (B.11)i,j,2 − (B.11)i,j+1,2 − (B.12)j,i,2 + (B.12)j,i+1,2
++ (B.13)i,j,2 − (B.13)i,j+1,2 + (B.21).
+(B.36)
+Next, let us compute [1 ⊗ Hi,1, Fj] − [1 ⊗ Hi,1, Fi]. By a direct computation, we obtain
+[1 ⊗ Hi,1, ℏ
+�
+w∈Z
+b
+�
+v=n+1
+ev,jt−w ⊗ ej,vtw]
+= − i
+2ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δi,jev,jt−w ⊗ ej,vtw + i
+2ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ej,vtw
+− ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δi,jev,jt−w ⊗ ej,vtwei+1,i+1 − ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ei,iej,vtw
++ ℏ2 �
+s≥0
+i
+�
+u=1
+�
+w∈Z
+b
+�
+v=n+1
+δi,jev,jt−w ⊗ ei,ut−seu,vts+w
++ ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j ≤ i)ev,jt−w ⊗ ei,vtw−sej,its
++ ℏ2 �
+s≥0
+a
+�
+u=i+1
+�
+w∈Z
+b
+�
+v=n+1
+δi,jev,jt−w ⊗ ei,ut−s−1eu,vts+w+1
++ ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j > i)ev,jt−w ⊗ ei,vtw−s−1ej,its+1
+− ℏ2 �
+s≥0
+i
+�
+u=1
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ei+1,ut−seu,vts+w
+− ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j ≤ i)ev,jt−w ⊗ ei+1,vtw−sej,i+1ts
+− ℏ2 �
+s≥0
+a
+�
+u=i+1
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ei+1,ut−s−1eu,vts+w+1
+− ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δ(j > i)ev,jt−w ⊗ ei+1,vtw−s−1ej,i+1ts+1.
+(B.37)
+By the assumption i < j, we obtain
+[1 ⊗ Hi,1, Fj] − [1 ⊗ Hj,1, Fi]
+= (B.37)j,i+1,1 + (B.37)i,j,2 + (B.37)j,i+1,3 + (B.37)i,j,4 + (B.37)j,i+1,5
+− (B.37)j,i,6 + (B.37)j,i+1,6 + (B.37)j,i+1,7 + (B.37)i,j,8 − (B.37)i,j+1,8
++ (B.37)i,j,9 − (B.37)j,i,10 + (B.37)j,i+1,10 + (B.37)i,j,11 + (B.37)i,j,12 − (B.37)i,j+1,12.
+By a direct computation, we obtain
+(B.37)j,i+1,1 + (B.37)i,j,2
+30
+
+= −j
+2ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δj,i+1ei+1,vtw ⊗ ev,i+1t−w + i
+2ℏ
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jej,vtw ⊗ ev,jt−w
+= −1
+2ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δj,i+1ei+1,vtw ⊗ ev,i+1t−w.
+(B.38)
+By a direct computation, we obtain
+(B.38) + (B.21) = 0.
+(B.39)
+By using
+(B.37)j,i+1,5 + (B.37)i,j,9
+= ℏ2 �
+s≥0
+j
+�
+u=1
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,i+1t−w ⊗ ej,ut−seu,vts+w
+− ℏ2 �
+s≥0
+i
+�
+u=1
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ei+1,ut−seu,vts+w
+= ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,i+1t−w ⊗ ej,jt−sej,vts+w
+(B.40)
+and
+(B.37)j,i+1,7 + (B.37)i,j,11
+= ℏ2 �
+s≥0
+n
+�
+u=j+1
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ei+1,ut−s−1eu,vts+w+1
+− ℏ2 �
+s≥0
+n
+�
+u=i+1
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ei+1,ut−s−1eu,vts+w+1
+= −ℏ2 �
+s≥0
+�
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,jt−w ⊗ ei+1,i+1t−s−1ei+1,vts+w+1.
+(B.41)
+we obtain
+(B.37)j,i+1,5 + (B.37)i,j,9 + (B.37)j,i+1,7 + (B.37)i,j,11
+= ℏ2 �
+w∈Z
+b
+�
+v=n+1
+δi+1,jev,i+1t−w ⊗ ej,jej,vtw.
+(B.42)
+By a direct computation, we obtain
+(B.42) + (B.19) = 0,
+(B.43)
+(B.37)j,i+1,3 + (B.9)j,i+1,3 = 0,
+(B.44)
+(B.37)i,j,4 + (B.9)i,j,1 = 0,
+(B.45)
+− (B.10)j,i,2 + (B.37)i,j,8 = 0,
+(B.46)
+(B.10)j,i+1,2 + (B.37)i,j,12 = 0,
+(B.47)
+(B.13)i,j,2 + (B.37)j,i+1,6 = 0,
+(B.48)
+− (B.13)i,j+1,2 + (B.37)j,i+1,10 = 0,
+(B.49)
+(B.11)i,j,2 − (B.37)j,i,6 = 0,
+(B.50)
+31
+
+− (B.11)i,j+1,2 − (B.37)j,i,10 = 0,
+(B.51)
+(B.12)j,i,2 + (B.37)i,j+1,8 = 0,
+(B.52)
+(B.12)j,i+1,2 + (B.37)i,j+1,12 = 0.
+(B.53)
+This completes the proof of Lemma B.6.
+Acknowledgement
+The author wishes to express his gratitude to Daniele Valeri. This article is inspired by his lecture
+at ”Quantum symmetries: Tensor categories, Topological quantum field theories, Vertex algebras”
+held at the University of Montreal. The author is also grateful to Thomas Creutzig for proposing
+this problem. The author expresses his sincere thanks to Nicolas Guay and Shigenori Nakatsuka
+for the useful advice and discussions.
+References
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+

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+Eur. Phys. J. C manuscript No.
+(will be inserted by the editor)
+The carbon star mystery: forty years later
+Theory and observations
+Oscar Straniero a,1, Carlos Abiab,2, Inma Dom´ınguezc,2
+1INAF - Osservatorio Astronomico d’Abruzzo, Via Maggini snc, I-64100 Teramo, Italy
+2Dpto. F´ısica Te´orica y del Cosmos, Universidad de Granada, E-18071 Granada, Spain
+Received: date / Accepted: date
+Abstract In 1981 Icko Iben Jr published a paper en-
+titled ”The carbon star mystery: why do the low mass
+ones become such, and where have all the high mass
+ones gone?”, where he discussed the discrepancy be-
+tween the theoretical expectation and its observational
+counterpart about the luminosity function of AGB car-
+bon stars. After more than 40 years, our understanding
+of this longstanding problem is greatly improved, also
+thanks to more refined stellar models and a growing
+amount of observational constraints. In this paper we
+review the state of the art of these studies and we briefly
+illustrate the future perspectives.
+Keywords Stellar evolution · Nucleosynthesis · Stellar
+abundances
+1 Introduction
+Carbon stars (or C stars) are those showing an abun-
+dance ratio by number C/O> 1. They were firstly iden-
+tified as a separate spectroscopic class by father Angelo
+Secchi. In a report to the French Academy of Sciences
+dated back to 1868, he wrote: ”...stars which do not
+belong to the three established types are very rare... I
+believe that they will belong to the family of the red
+stars and of variable stars”. For a long time, the ori-
+gin of this peculiar stellar chemistry has been a mys-
+tery [1]. A carbon enhancement can be produced ei-
+ther by an internal mixing of freshly synthesised car-
+bon (intrinsic carbon stars), or through the accretion
+of carbon-rich matter from a companion star in a bi-
+nary system (extrinsic carbon stars). As a consequence,
+ae-mail:oscar.straniero@inaf.it
+bcabia@ugr.es
+cinma@ugr.es
+the carbon enhancement may appear at different stages
+in the evolution of the stars, and, hence, a variety of
+carbon star spectral types are possible, depending on
+the actual effective temperature and gravity, and on
+the abundances of the molecules bearing carbon atoms
+(CN, C2, CH, see [2], [3]). In this paper we will focus
+on the so-named normal (N-type) carbon stars, i.e. the
+intrinsic carbon stars that form during the asymptotic
+giant branch (AGB) phase. In general, these C stars
+are believed to be important contributors to the carbon
+budget in the Galaxy. However, their net contribution
+is not yet clear, also in comparisons with other possible
+C polluters, such as massive Wolf-Rayet stars. To set-
+tle the question and quantify the relative contribution
+of AGB stars to the evolution of the galactic carbon
+abundance, it is necessary to know how much carbon is
+produced and ejected as a function of the initial mass
+and metallicity. The theory of stellar evolution teaches
+us that surface carbon enrichment in the AGB phase is
+a consequence of periodic episodes of convective mix-
+ing, named the third dredge-up (TDU), which trans-
+port material that has suffered He-burning up to the
+stellar surface [4], [5]. In practice, an AGB star under-
+goes recurrent thin-shell instabilities [6], called thermal
+pulses (TP), which induce thermonuclear runaways, or
+He-shell flashes, whose power may attain a few 108 L⊙
+(in AGB stars with M∼ 2 M⊙). A TDU episode may
+occur after a TP, when the external layers expand and
+cool down, until the H-burning shell eventually dies out,
+and the external convection can penetrate the He- and
+C-rich mantle. However, more than 40 years after the
+pioneering Iben’s paper on The Carbon star mystery [1],
+the efficiency of the TDU and the chemical yields from
+AGB stars are still burdened by heavy uncertainties and
+disagreements among different authors, mainly due to
+the lack of a robust theory of convection and mass loss.
+arXiv:2301.03978v1  [astro-ph.SR]  10 Jan 2023
+
+2
+An homogeneous and accurate set of spectroscopic and
+photometric observations could compensate such a the-
+oretical drawback.
+Carbon stars are also among the main sites where heavy
+elements (A≥ 90) are produced trough the slow capture
+of neutrons: the s process. Neutrons for this process are
+provided by the 13C(α, n)16O reaction, which is active
+at relatively low temperature (T ∼ 90 MK) during the
+period of time that elapses between two TPs (inter-
+pulse phase). According to the current paradigm, the
+partial mixing occurring at the bottom of the convective
+envelope at the time of the TDU leaves a thin pocket
+where the H mass fraction is XH < 0.01, while the
+carbon mass fraction is about 0.2. Then, at the H re-
+ignition, a substantial amount of 13C is produced by the
+12C(p, γ)13N reaction followed by the 13N decay, and,
+later on, the s-process can start, due to the activation
+of the 13C neutron source. A second neutron burst, as
+due to the activation of the 22Ne(α, n)25Mg reaction,
+may eventually occur at the bottom of the convective
+shell powered by a TP, but only if the temperature ex-
+ceeds 300 MK (see e.g. [7], and references therein). As
+a matter of fact, the s-process enhancement observed
+in normal C stars is mainly due to the 13C neutron
+burst, while the 22Ne source only provides a marginal
+contribution. The presence of Tc alive (99Tc half-life
+2.11 × 105 yr) in the atmosphere of C stars is a probe
+of their intrinsic nature.
+Carbon stars are also the parents of main stream SiC
+grains that may form in their cool and C-rich circum-
+stellar envelopes. Some meteorites that hit the Earth
+contain these stardust grains, which are isolated and
+analysed in the laboratory. In this way, SiC grains pro-
+vide valuable information on the physical conditions oc-
+curring in the circumstellar envelopes of C stars as well
+as on the internal nucleosynthesis processes [8].
+During the last few years a number of theoretical and
+observational studies shed new light on the scenario de-
+scribed above. Here we discuss some of these advances,
+as obtained by combining new theoretical models and
+more accurate observational constraints. In section 2
+we illustrate state-of-the-art models of C-star progeni-
+tors and their nucleosynthesis. In Section 3 we discuss
+recent observational advances and the new issues that
+these observations have revealed, and in Section 4, we
+summarise the current status and future prospects of
+this subject.
+2 Theory & models
+As discussed by I. Iben in his a seminal pape [1], the
+C-star luminosity function (N-type) of the Milky Way
+and of the Magellanic Clouds are peaked at Mbol ∼ −5,
+and very few C stars are brighter than Mbol ∼ −6 (see
+Sect. 3). This implies that i) the majority of the C stars
+should have mass between 1.5 and 2.5 M⊙, and ii) very
+rare C stars are observed whose mass exceeds 3-4 M⊙.
+Since 1981, many progresses have been done in mod-
+elling AGB stars and an answer to these questions have
+been partially found. Nevertheless, a general consensus
+has not yet been reached, because of the many uncer-
+tainties still affecting AGB stellar models. First of all,
+let us discuss the widely accepted scenario for the C-
+star formation.
+As it is well known, a TP-AGB stars is made of three
+zones, namely: a C-O-rich core, sustained by the pres-
+sure of degenerate electrons and cooled by the release
+of plasma neutrinos; an intermediate He-rich region,
+where recurrent TPs powered by He burning take place;
+and a H-rich envelope efficiently mixed by a rather
+deep convective envelope. For most of the time, the He-
+burning shell is off, and the luminosity is powered by
+the CNO bi-cycle active in a thin H-burning shell. The
+compression and heating of the matter left behind by H
+burning causes the He ignition and a convective shell,
+which extends over almost the entire He-rich zone, de-
+velops. In this way, the carbon produced by the triple-
+α reaction is mixed-up to the top of the He-rich re-
+gion. At the end of the TP, the carbon mass fraction
+in the most external layers of this intermediate zone
+is raised up to ∼ 20 %. So far, the presence of an ac-
+tive H-burning shell has maintained an entropy barrier
+that prevents the penetration of the external convec-
+tion into the underlying H-exhausted region. However,
+due to the outgoing energy flow generated by He burn-
+ing, the envelope expands and cools, until H burning
+dies out. In this condition, the external convection can
+penetrate the H-He discontinuity and, eventually, can
+reach the C-enhanced zone. In low-mass AGB stars, this
+deep mixing episode may occur a few hundred years af-
+ter the TP quenching, while in a massive AGB star
+it requires a much shorter time. Anyway, the resulting
+carbon dredge-up is the process responsible for the for-
+mation of an intrinsic C star. Depending on the initial
+metallicity, several TDU episodes may be required until
+the C/O> 1 condition is attained at the stellar surface.
+So, the question is: in which stars are the TDUs suffi-
+ciently intense to allow them to become C stars? In this
+context, we have understood that the efficiency of the
+dredge-up process depends on the concurrent actions
+of several stellar parameters, such as the core and the
+envelope masses, as well as the initial metallicity (see,
+e.g., [9]). In the following we try to disentangle the vari-
+ous physical processes that affect the carbon dredge-up
+in AGB stars.
+
+3
+2.1 The shell H-burning
+The TDU is the result of the expansion and cooling
+of the envelope that follows the violent He-ignition.
+This occurrence causes the temporary stop of H burn-
+ing and an increase of the radiative opacity, and both of
+these phenomena favours the penetration of the exter-
+nal convective zone into the H-exhausted region. It goes
+without saying that the ultimate engine of the TDU is
+the He-burning thermonuclear runaway. As a matter
+of fact, deeper TDUs are found after stronger thermal
+pulses1.
+On the other hand, the H-burning rate during the inter-
+pulse period determines the He-ignition conditions and,
+in turn, the strength of the TP. In particular, the He-
+ignition density is larger in case of slower H burning
+and, in turn, a higher peak luminosity is attained dur-
+ing the thermal runaway. For instance, [10] (see also
+[11]) have shown how a reduction of the 14N(p, γ)15O
+reaction rate leads to stronger TPs and deeper TDU
+episodes. Indeed, this reaction is the bottleneck of the
+CNO and its rate controls the rate of the shell H burn-
+ing. On the other hand, the H-burning efficiency also
+depends on the mass of the H-exhausted core [12] [9].
+Hence, stronger TPs and, in turn, deeper TDUs are
+found in low-mass AGB stars, those with a smaller core
+mass and, in turn, a less efficient shell H burning. In
+principle, also the initial metallicity, more precisely the
+initial abundances of C, N, and O, affects the strength
+of the first few TPs: the lower the CNO abundance the
+stronger the thermal pulses and the deeper the TDU.
+However, in the late part of the AGB, the CNO abun-
+dances in the envelope are modified by the TDUs and
+the influence of the initial metallicity disappears.
+2.2 The hot-bottom-burning (massive AGB stars only)
+During the inter-pulse periods of massive AGB stars
+(M ≥ 4 M⊙), the bottom of the convective envelope
+penetrates the zone where H burning is active. This
+phenomenon, which makes massive AGB stars impor-
+tant sites for the nucleosynthesis of various light and
+intermediate mass isotopes, is known as hot-bottom-
+burning (HBB). The temperature of the deeper layer of
+the convective envelope may be ∼ 30×106 K in a 4 M⊙
+(solar composition) and up to 100 × 106 K in a 7 M⊙
+star. A lower metallicity favour the HBB, because of the
+less steep entropy barrier at the H-burning shell, which
+is, for this reason, more easily penetrated by convec-
+tive instability. This phenomenon has two major conse-
+1The strength of a thermal pulse may be measured by the
+maximum luminosity attained by He-burning during a TP.
+quences. Firstly, fresh H is brought into the H-burning
+layers. As a result, the shell H-burning is more efficient
+and, in turn, the TPs are weaker and the TDUs are
+shallower. In addition, carbon, primordial or carried in
+the envelope by the TDU, is mostly transformed into ni-
+trogen through the CN cycle active at the bottom of the
+external convective region. So, massive AGB stars are
+expected to become N-rich instead of C-rich. This pro-
+cess eventually ceases when the envelope mass, which is
+eroded by mass loss, reduces down to ∼ 2.2 M⊙. There-
+fore, approaching the AGB tip, a star with mass larger
+than 4 M⊙ may still become C-rich, but just for a short
+time.
+2.3 The hot third dredge-up (massive AGB stars only)
+As previously said, the TDU starts just after a TP,
+when H burning dies out. In massive AGB stars, how-
+ever, when the convective instability penetrates the H-
+exhausted core, it encounters layers where the temper-
+ature is sufficiently high to re-activate proton-capture
+reactions. Then, the energy released by nuclear reac-
+tions contrasts the convective instability that is pushed
+outward, thus causing a premature stop of the TDU.
+This phenomenon, called hot third dredge-up (HTDU)
+significantly limits the TDU in massive AGB stars [13].
+In passing, let us note that owing to the HTDU, the
+s-process yields from the more massive AGB stars are
+expected to be negligible.
+2.4 The mass-loss
+The AGB phase terminates when the mass-loss erodes
+the envelope until H burning is substantially suppressed.
+For a 2 M⊙ (solar composition), the star is expected to
+leave the AGB when the envelope mass is reduced down
+to ∼ 0.1 M⊙, but the TDUs become progressively shal-
+lower when the envelope mass becomes lower than ∼ 0.5
+M⊙ (see, e.g., [14]. In this context, the AGB mass-loss
+rate determines the number of TDU episodes and the
+total amount of carbon that is dredged-up during the
+AGB phase. In other words, the mass-loss rate deter-
+mines the possibility for an AGB star of becoming a C
+star or not. In addition, for stars with mass lower than
+∼ 2 M⊙ also the pre-AGB mass-loss play an important
+role. When these stars leave the main sequence, they
+enter the RGB phase, during which they lose up to a
+few tenths of solar masses. Then, these stars approach
+the TP-AGB phase with an already eroded envelope.
+
+4
+2.5 Boundary mixing and extra-mixing
+When the convective envelope penetrates the H-exhausted
+zone, a sharp variation of the composition takes place
+at the convective boundary. In less than 10−3 M⊙, the
+H mass-fraction drops from about 0.7 to 0. Owing to
+this composition discontinuity, a sharp variation of the
+radiative opacity, associated to an abrupt change of
+the radiative gradient, develops. In these conditions,
+the precise location of the convective border (i.e. the
+limit of the region fully mixed by convection) becomes
+highly uncertain. An initially small perturbation caus-
+ing a mixing just below the convective boundary is
+amplified on a dynamical timescale, so that the ra-
+diative gradient in the radiative stable zone rises up
+and the convective instability moves inward. This con-
+dition is commonly encountered in stellar model compu-
+tations at the time of the second and the third dredge-
+up (see e.g., [15], [16], [17], [18], [14]). While the ef-
+fect of such an instability is marginal in the case of
+the second dredge-up [18], the deepness of the TDU is
+significantly extended [14]. In order to correctly treat
+this phenomenon, a more realistic description of the
+convective boundary than that usually adopted in ex-
+tant stellar evolution codes is required. Instead of a
+well defined spherical surface, as obtained when the
+bare Schwarzschild’s criterion is used, the transition be-
+tween the full-radiative core (i.e. unmixed) and the full-
+convective (i.e. fully mixed) envelope likely occurs in an
+extended zone where only a partial mixing takes place
+(semi-convective layer), so that a smooth and stable
+H-profile may form (see Figure 1). Within this transi-
+tion zone, the convective velocity smoothly drops from
+about 105 cm/s, at the convective boundary, to 0. Note
+that this process may also solve another longstand-
+ing issue of AGB stars, that is the formation of the
+13C-pocket needed to activate a substantial s-process
+nucleosynthesis during the inter-pulse phase (see, e.g.,
+[14]). It is indeed in this transition zone left after a
+TDU episode that a suitable amount of 13C may form,
+through the 12C(p.γ)13N reaction followed by the 13N
+decay. In spite of the many efforts done to incorporate
+this phenomenon in one-dimension hydrostatic stellar
+evolution codes, the evaluation of the actual extension
+of this transition zone and the degree of mixing there
+would require more sophisticated tools. In any case, the
+larger the extension of this transition zone the deeper
+the resulting TDU (see, e.g., [9]).
+In addition to convection, other processes causing mix-
+ing below the convective envelope may affect the TDU
+and the formation of the 13C-pocket. Rotational in-
+duced instabilities were early considered [20], [21]. Ac-
+cording to the extant models, mixing induced by rota-
+convective velocity
+Pressure
+Fig. 1 The boundary of the convective envelope during the
+TDU for a 2 M⊙ with solar composition. Upper panel: chem-
+ical composition in the transition region between the convec-
+tive envelope and the radiative He-rich zone. Lower panel: the
+exponential decline of the convective velocity and the pres-
+sure gradient. Adapted from [19].
+tion produces a marginal effect on the TDU, while it
+could modify the 13C-pocket, after its formation, and
+the consequent s-process nucleosynthesis. Nevertheless,
+recent asteroseismic studies of evolved low-mass stars
+revealed that most of the internal angular momentum
+is lost before the AGB phase, so that rotation likely
+does not play a relevant role in the AGB evolution and
+nucleosynthesis (see, e.g., [22] and references therein).
+More promising is the hypothesis of mixing induced by
+internal gravity wave (IGW) generated at the boundary
+of the convective envelope [23]. The connection between
+internal convective zone and IGWs clearly emerges in
+various hydrodynamic simulations. This expectation is
+confirmed by the detection of g-mode (low-frequency)
+variability in photometry studies of main-sequence stars
+with convective cores [24]. Likely, this process could also
+induce some mixing below the boundary of the con-
+vective envelope of AGB stars. The persistence of an
+internal magnetic field could also generate mixing in
+the He-rich zone, through the so-called magnetic buoy-
+ancy [25]. [26] have recently investigated the effect of
+this mechanism in low-mass AGB stars and conclude
+that the resulting s-process nucleosynthesis is in bet-
+ter agreement with the abundance patterns observed in
+AGB stars and with the isotopic composition of C-rich
+pre-solar grains that are supposed to originate in the
+cool atmosphere of C stars. However, the actual effect
+
+5
+of all these (non-convective) mixing processes on the
+TDU effciency has not be clearly established yet.
+2.6 Predictions of extant AGB models
+Let us finally come back to to the C-star mystery. Al-
+though a reliable evaluation of which stars may become
+C-rich before leaving the AGB is still hampered by un-
+certainties on AGB mass loss and boundary mixing,
+extant stellar models provide a coherent, even if quali-
+tative, picture.
+In Figure 2, we report the minimum and the maxi-
+mum mass of stars expected to become C-rich dur-
+ing the AGB phase as a function of the metallicity.
+The minimum masses are from the FRUITY database
+(http://fruity.oa-teramo.inaf.it/) and were calculated by
+means of the FuNS code [4]. In particular, the AGB
+mass-loss rate was calculated by means of an empirical
+mass-loss vs period relation while the treatment of the
+boundary mixing is based on an exponential decay of
+the convective velocity below the convective envelope.
+The latter is a quite trivial consequence of the pen-
+etration of convective bubbles, which are accelerated
+in the convective envelope, into the underlying stable
+zone. When a bubble penetrate the stable zone, it is
+decelerated by the buoyancy at a rate:
+¨r = −αv2
+(1)
+where r is the distance from the internal border of the
+convective envelope and the α parameter has the di-
+mension of the inverse of a distance. Hence, if v0 is the
+velocity at r = 0 (convective boundary), after a time
+integration, one may easily find:
+v0
+v = αv0t + 1
+(2)
+and, after a further integration:
+−r = 1
+α ln(αv0t + 1)
+(3)
+finally, by means of equation 2:
+v = v0 exp(−αr) = v0 exp(−
+r
+βHP
+)
+(4)
+where β = 1/αHP is a free dimensionless parameter
+and HP is the pressure scale-height. Note the similar-
+ity of the velocity gradient with the pressure gradient
+(except for the sign). Indeed, according to the hydro-
+static equilibrium equation, the pressure gradient can
+be written as:
+P = P0 exp( r
+HP
+)
+(5)
+(both r and P increase toward the centre). The result-
+ing velocity and pressure within the convective bound-
+ary layer of a 2 M⊙ model (solar initial composition),
+0
+1
+2
+3
+4
+5
+6
+0.0000
+0.0050
+0.0100
+0.0150
+0.0200
+mass (Mꙩ)
+Z
+C/O>1
+C/O<1
+C/O<1
+Fig. 2 Minimum and maximum initial mass for C-star pro-
+genitors versus metallicity, according to the theoretical pre-
+dictions we obtain by means of the FuNS code (see text).
+‐7.5
+‐7
+‐6.5
+‐6
+‐5.5
+‐5
+‐4.5
+‐4
+‐3.5
+0.0000
+0.0050
+0.0100
+0.0150
+0.0200
+Mbol
+Z
+C/O>1
+C/O<1
+C/O<1
+Fig. 3 Minimum and maximum bolometric magnitude for C
+stars versus metallicity, according to the theoretical predic-
+tions we obtain by means of the FuNS code (see text).
+during a TDU episode, are shown in Figure 1 (arranged
+from [4]). This simple argument is confirmed by more
+sophisticated hydrodynamic simulations [27]. If some
+extra-mixing process is also at work, such as that due
+to IGWs or to magnetic buoyancy, the velocity profile
+within the boundary layer may be modified. In recent
+works, a convolution of two exponential decay functions
+is adopted in order to mimic this occurrence [28].
+The maximum masses in Figure 2 have been instead
+obtained by means of a more recent version of the FuNS
+code, in which a more appropriate numerical scheme to
+treat the HBB and HTDU is adopted. In particular,
+the differential equations describing the stellar struc-
+ture in hydrostatic and thermal equilibrium are fully
+coupled to the differential equations describing the evo-
+lution of the internal composition, as due to mixing and
+thermonuclear burning processes. The qualitative pic-
+ture is clear. The minimum mass is limited by the mass
+
+6
+loss efficiency during both the pre-AGB and the AGB
+phases. It is lower at lower Z because less TDU episodes
+are needed before attaining the condition C/O> 1. The
+maximum mass depends on the AGB mass-loss rate and
+it is limited by the onset of the HBB and the HTDU
+phenomena. In this case, a lower initial O abundance
+allows the formation of C stars with higher masses. Sim-
+ilarly, in Figure 3 we show the corresponding maximum
+and minimum bolometric magnitude of C stars. Note-
+worthy, these results coupled to the evolutionary time
+spend by each C-star progenitor up to the AGB phase
+and the duration of the C-star phase, are the basic
+ingredients to construct theoretical C-star luminosity
+functions, but other ingredients are needed, such as the
+star formation history, the initial mass function and the
+metallicity vs age relation. As pointed out by [29] (see
+also [30]), the luminosity function spread may be sub-
+stantially affected by these additional parameters.
+3 The observational framework
+Normal carbon stars represent a formidable challenge
+from the spectroscopic point of view. They show very
+crowded spectra due to their low temperatures (Teff ∼
+3000 K) and strong molecular absorptions. Further-
+more, most of AGB carbon stars are variable, thus their
+spectra are usually affected by large scale movements
+of the photosphere (stellar pulsations, shock waves...),
+which provoke strong line asymmetries, broadening and
+Doppler shifts. These phenomena greatly hinders the
+chemical analysis of these stars, and in principle would
+require the use of dynamical atmosphere models. Al-
+though some progress has been made in this sense [31],
+still most of the chemical analysis rely on the basis of
+static atmosphere models assuming LTE. This may in-
+troduce systematic errors in the determination of their
+chemical, and obviously, lead to wrong conclusions on
+the nucleosynthetic processes occurring in their inte-
+riors. Despite of this, considerable progress has been
+achieved in the past few decades and the abundance
+analyses show a comfortable agreement with theoret-
+ical predictions, in particular concerning the observed
+abundances of Li, F, C, N, O (and their isotopic ra-
+tios), and s-process elements. Next, we summarise the
+main achievements and the new issues that these abun-
+dance analyses have revealed. We will not discuss here
+the signi���cant observational advances performed on the
+formation and structure of the circumstellar envelopes
+of carbon stars and their implications on the dust for-
+mation and the mass-loss rate history.
+Carbon stars show only a few spectral windows (e.g.
+λ ∼ 4800 − 5000 ˚A, or λ ∼ 7700 − 8100 ˚A) suit-
+able for abundance analysis at optical wavelengths pro-
+vided that CN, C2, CH and other C-bearing molecu-
+lar features are included in any spectroscopic line list.
+In this sense, a significant effort has been made in the
+last years to improve the wavelength positions, energy
+levels and line intensities for all the isotopic combi-
+nations of the above mentioned molecules [32]. The
+near infrared (NIR) spectrum of N-type stars usually
+is less crowded allowing the identification of interest-
+ing atomic and molecular features, which makes the
+abundance analysis less difficult although still has to
+be explored with detail since many spectroscopic fea-
+tures (probably of atomic nature) are unidentified [33].
+In any case, to perform an accurate abundance analysis
+in these stars the use very high resolution spectroscopy
+is mandatory, and unfortunately, still few high reso-
+lution NIR spectrographs attached to medium and/or
+large-size telescopes are available.
+3.1 The C/O and 12C/13C ratios
+As mentioned in Sect. 2, carbon stars occupy the tip
+in the AGB spectral sequence M→MS→S→SC→C(N),
+thus they are the natural result of the continuous mix-
+ing of carbon into the envelope throughout the TDU
+after each TP. As a consequence the C/O ratio is ex-
+pected to increase continuously along the AGB phase
+until mass loss terminates the evolution. Surprisingly,
+the derived C/O ratios so far do not greatly exceed
+unity (∼ 1.0 − 1.5, e.g. [34], [35]). Only a few metal-
+poor N-type stars, observed in metal-poor extragalac-
+tic stellar systems, show significantly higher C/O ratios
+(4-8, [36], [37]). Although this is in agreement with the
+fact that the formation of a carbon star is easier at low
+metallicity because of the lower O content in the en-
+velope and the increase of the efficiency of the TDU,
+the C/O ratios derived in the overwhelming majority
+of carbon stars are still considerably lower than theo-
+retical predictions. Depending on the initial mass and
+metallicity, C/O ratios larger than 10 are predicted at
+the end of the AGB. It has been suggested that the
+last part of the AGB evolution, is occupied by infrared
+extremely carbon-rich objects enshrouded in a thick
+dusty envelope, so that photospheric abundances are
+not accessible. Also, as carbon exceeds oxygen in the
+envelope, it may condense into grains removing carbon
+atoms from the gas phase and, as consequence, keeping
+the C/O ratio only slightly larger than unity. In any
+case, this question remains unsolved. Note that most of
+post-AGB stars show also C/O ratios very close to 1
+[38], which is not either understandable on theoretical
+grounds.
+Another issue directly related with the actual C/O
+ratio in the envelope of carbon stars is the 12C/13C ra-
+
+7
+tio. [34,51] identified a significant number (∼ 25 %) of
+normal C stars with 12C/13C∼ 10 − 30, while the aver-
+age observed ratio is ∼ 70 [33]; this latter value agrees
+with theoretical expectations when C/O exceeds unity
+in the envelope during the AGB phase. Indeed, this iso-
+topic ratio is expected to increase continuously as 12C
+is being added into the envelope by the TDU to val-
+ues larger than 100-200. Anomaly low carbon isotopic
+ratios are observed also in many low-mass RGB stars.
+Several authors (e.g. [39]) have shown that this chem-
+ical anomaly in RGB stars may derive from transport
+mechanisms linking the envelope to zones where par-
+tial H-burning occurs. This phenomena is called “extra-
+mixing” or ”deep mixing”, and its nature is still highly
+debated [40]. [41] suggested that a similar mixing mech-
+anism would operate during the AGB in order to ex-
+plain the low 12C/13C values observed in some N-type
+stars. These authors show that, even assuming a nor-
+mal 12C/13C value (∼ 20) at the end of the RGB phase,
+ratios larger than ∼ 40 are unavoidable reached when
+C/O= 1 at the AGB phase. The operation of this mix-
+ing mechanism during the AGB phase would be, never-
+theless, rather tricky since only for a few carbon stars
+16O/17O/18O ratios compatible with the operation of
+such a mechanism are found, although their observed
+12C/13C and 14N/15N ratios would be difficult to rec-
+oncile within this scenario [32],[42]. Note however, that
+mainstream SiC grains, formed in the envelopes of N-
+type stars, provide the same evidence: in the St. Louis
+database, for 23% of them, the carbon isotope ratio is
+below 40, while the average value is around 70, as ob-
+served in normal carbon stars.
+Observational evidence favouring the existence of
+non-standard mixing mechanism(s) in the AGB phase
+is provided by the Li observations. Around 2 − 3 % of
+galactic N-type stars show Li enhancements; a few show
+a huge 6708 ˚A LiI absorption line; these stars are super
+Li-rich (A(Li)≥ 4.0, [43]). In fact AGB carbon stars are
+believed to be significant contributors to the Li bud-
+get in the Galaxy. Li can be produced in AGB stars
+by the operation of the Cameron & Fowler mechanism
+[44] at the bottom of a moderately hot (T> 3.0 × 107
+K) convective envelope. But, as mentioned in the pre-
+vious section, these temperatures are reached in lumi-
+nous (Mbol ≤ −5.0 mag) AGB stars with M≥ 4 − 5
+M⊙ where, in addition, the proton captures on carbon
+at the bottom of the convective envelope may prevent
+the formation of a carbon star (see Fig. 2). In fact Li-
+enhancements are found in very luminous O-rich AGB
+stars [45],[46]. However, the fact that the luminosity
+function of N-type stars indicates that the overwhelm-
+ing majority of them are of low-mass (M≤ 3 M⊙) (see
+below), seems to discard the HBB as the mechanism
+responsible of the Li production in carbon stars. Note
+that some of the Li-rich N-type stars are also 13C-rich.
+Similarly to the 12C/13C issue, it has been suggested
+an explanation in terms of deep mixing [47].
+3.2 Fluorine
+The source of fluorine in the Universe is currently widely
+debated, and several sites have been proposed as poten-
+tial candidates. However, only in AGB and post-AGB
+stars there is a direct observation of fluorine produc-
+tion provided by spectroscopic findings of photospheric
+[F/Fe]2 enhancements [48], [49]. Fluorine can be pro-
+duced during the AGB phase through an intricate nu-
+clear chain in such a way that its envelope abundance is
+expected to be correlated with the abundances of car-
+bon and s-process elements. In fact this nuclear chain
+confers to this element both a primary and secondary
+origin. Recent F abundance determinations from HF
+lines at 2.3 µm in Galactic and extragalactic AGB car-
+bon stars have confirmed large [F/Fe] enhancements as
+well as an increasing trend of this enhancement with
+the decreasing metallicity [49]. However, while obser-
+vations and theory agree at close-to-solar metallicity,
+stellar models at lower metallicities overestimate the
+fluorine production, in particular the abundance ratio
+between F and s-elements, which are also produced in
+AGB carbon stars. This discrepancy has lead to modify
+the driving process for the formation of the 13C-pocket
+with respect to the standard parameterisation (see Sect.
+2). Recent AGB stellar models with mixing induced by
+magnetic buoyancy at the base of the convective enve-
+lope agree much better with available fluorine spectro-
+scopic measurements at low and close-to-solar metal-
+licity [26]. However, when the computed AGB fluorine
+yields are introduced in a galactic chemical evolution
+model, it becomes evident that other fluorine sources
+than AGB stars are required [50].
+3.3 s-elements
+It was [51] who firstly reported the enhancements of
+s-elements in the surface of carbon stars. This author
+found that N-type stars were typically of solar metal-
+licity, presenting mean s-process element enhancements
+of a factor of 10 with respect to the Sun. However,
+2We adopt the usual notation [X/Y]≡ log (X/Y) − log
+(X/Y)⊙ for the stellar value of any abundance ratio X/Y
+(by number). In the following we use ‘ls’ to refer to the light
+mass s-elements Y and Zr, and ‘hs’ to denote the high mass
+s-elements Ba, Nd, La and Sm.
+
+8
+more accurate studies, based on higher resolution spec-
+tra and better analysis tools [34], [35], [36], [52], have led
+to strong revisions in the quantitative s-element abun-
+dances. N-type stars were confirmed to be of near so-
+lar metallicity, but they show on average <[ls/Fe]>=
++0.67 ± 0.10 and <[hs/Fe]>= +0.52 ± 0.29, which is
+significantly lower than that estimated by [51]. These
+values are of the same order as those derived in the
+O-rich S stars [53]. From these observations two main
+conclusions are reached: a) The abundance ratio be-
+tween Rb and its neighbours (Sr, Y, Zr) indicates that
+the main neutron source operating in AGB stars is the
+13C(α, n)16O reaction and, therefore, than carbon stars
+are of low-mass (≤ 3 M⊙); b) the [hs/ls] ratio (i.e.
+the abundance ratio between the heavy (Ba, La, Ce)
+and light (Sr, Y, Zr) s-elements), a parameter sensitive
+to the neutron exposure, increases with the decreas-
+ing metallicity of the star in agreement with theoretical
+predictions of the s-process. However, this ratio shows
+a significant dispersion at a given metallicity, which is
+usually interpreted as a signature of that existing in the
+13C-pocket (abundance mass fraction profile, amount of
+13C burnt) in AGB stars. This dispersion, on the other
+hand, is necessary to account for the s-element abun-
+dance patterns observed in individual C-stars. We note,
+nevertheless, that the correlation [hs/ls] vs. [Fe/H] is
+not clearly observed in post-AGB stars, the progeny of
+AGB stars, at least in the metallicity range studied [38].
+This shows the complexity of the s-process nucleosyn-
+thesis in AGB stars.
+3.4 Luminosity function
+Finally, the release of the Gaia DR3 catalogue made it
+possible to determine accurate distances (and hence lu-
+minosities) to the Galactic AGB carbon stars and con-
+strain their positions in the HR diagram, their Galactic
+location and population membership. Figure 4 shows
+the luminosity function derived in a sample of ∼ 300
+Galactic carbon stars (N-type) with parallax accuracy
+better than 10% according to Gaia DR3 [54]. The av-
+erage luminosity is Mbol = −5.04 ± 0.55 mag, slightly
+brighter than the average luminosity derived in carbon
+stars in the Magellanic Clouds. This figure agrees with
+theoretical expectations that C stars are formed more
+easily at low metallicities, thus earlier during the AGB
+phase (lower luminosity, see previous Section). How-
+ever, Fig. 4 shows the existence of significant luminos-
+ity tails both at low and high Mbol values at which
+theoretically carbon stars would not exist because this
+would imply a progenitor mass low-er/higher than the
+corresponding limits for their formation. Note however,
+as mentioned in Sect. 2, that other factors may affect
+the spread of the luminosity function. Nevertheless, the
+low luminosity tail (Mbol ≥ −4.0) can be explained if
+a small fraction of the stars are extrinsic, i.e. they are
+low-mass stars (< 1.5 M⊙) that become C-rich because
+the accretion of carbon rich material in a binary system
+and then enter the AGB phase already with C/O> 1
+in the envelope. Other possibility is that the mass limit
+for the operation of efficient TDU and, thus, for the for-
+mation of a carbon star is lower than expected (see Fig.
+2). Some observational evidence of this latter hypothe-
+sis exists [55]. The high luminosity tail (Mbol ≤ −5.5)
+is more difficult to explain since these luminosities are
+attained by intermediate mass stars (M> 4 − 5 M⊙),
+which theoretically will not become a C star (at least
+with near solar metallicity, see Fig. 3) because the oper-
+ation of the HBB. The existence of these high luminos-
+ity carbon stars (a few have been also observed in the
+Magellanic Clouds), implies that our understanding of
+the formation of a C star is still incomplete (see Sect.
+2).
+4 A final remark
+There are also observational evidence of carbon enrich-
+ment occurring before the TP-AGB phase. These are
+the so called R-hot type carbon stars, with near so-
+lar metallicity and no s-element enhancement [56,57].
+Many of them have luminosities compatible with red
+clump stars (central He burning) [58,59]. The evolu-
+tionary phase that could explain the needed mixing may
+well be the He-flash, provided that He is ignited at the
+border of the He-core, thus close to the H-shell, and
+at high degenerate physical conditions [60]. It was sug-
+gested that rotation may lead to this off-center ignition
+[56,61]. Moreover, as no R stars were found in binary
+systems, which statistically is unlikely, [62] suggested
+that they originate from binary mergers.
+No consistent evolutionary scenario has been found
+so far to explain these stars. One of the most popu-
+lar, in terms of population synthesis analysis, is the
+merger of a He WD with a RGB star [63]. After the
+merger, the resulting star evolves and, eventually, a de-
+generate He ignition occurs followed by a deep carbon
+dredge-up. [64] firstly investigated this scenario. Based
+on three-dimension SPH simulations of the merger of
+a He-WD with different masses and a RGB star, and
+one-dimension hydrostatic simulations of the accretion
+phase and the evolution up to the HB phase, they show
+that the dredge-up of freshly synthesised carbon does
+not occur in any of their models. This includes mas-
+sive He WDs, that were found to be good candidates
+by [65,66]. In contrast, for massive He WDs, [64] ob-
+tained that if He is ignited after the accretion phase,
+
+9
+the He-flash is mild as the physical conditions at the
+border of the He-core are not highly degenerate. In this
+case, the convective He-shell remains confined inside
+the He-rich region and, later on, the entropy barrier
+due to the active H-burning shell, prevents any mixing.
+On the other hand, if He ignites during the accretion
+phase, the accretion disk prevents any penetration of
+the convective envelope. However, all these calculations
+of He flash models were done with a one dimension hy-
+drostatic code and this occurrence may likely be their
+major limit. [67] argue that to properly treat convec-
+tion, a three-dimension hydrodynamical model should
+be preferred. However, this type of numerical simula-
+tions only covers about 1 day of the stellar evolution.
+The models performed by [67] show the growth of the
+He-convective unstable zone toward the H-rich layers on
+a dynamical timescale; some mixing could occur, but it
+does not take place during the simulation. Thus, the
+question remains open.
+Recently, by analysing a large sample of Galactic
+carbon stars for which very accurate astrometry is avail-
+able from Gaia DR3, [46] found many R-hot stars with
+low luminosities, covering all the RGB phase. This shows
+that, at least for some R-hot stars, carbon enrichment
+should occur before the He-flash. In this framework, an
+extrinsic origin appears favoured.
+5 Future perspectives
+A bright future is expected for studies on both normal
+and R-type C stars. The detection of isotopic abun-
+dances is a key tool to understand the interplay be-
+tween mixing and burning processes in stars. However,
+the optical and near-IR spectrum of a C star is a for-
+est of closely-spaced absorption lines. Thus, very high
+spectral resolution, coupled to a high signal-to-noise ra-
+tio, is required to distinguish lines of different isotopes.
+As a matter of fact, only some isotopic abundances of
+C and O have been measured so far. In this context,
+the next generation of large aperture telescopes (ELT
+and its competitors) will provide a great opportunity
+to improve our understanding of C stars. For instance,
+the high-resolution ELT instrument ANDES, formerly
+known as HIRES, will deliver high-quality stellar spec-
+tra (R> 100, 000 and S/N> 100) of AGB stars belong-
+ing to the Local Group of Galaxies [68]. This obser-
+vational effort must be necessarily accompanied by the
+identification and improvement of the spectroscopic pa-
+rameters of C- and O-bearing molecular lines in the vi-
+sual and infrared wavelength ranges.
+On the other hand, asteroseismic studies can provide an
+unprecedented view on the internal structure of stars
+in different phases and their evolution. So far, data
+Fig. 4 Luminosity function for about 300 Galactic AGB car-
+bon stars derived from Gaia DR3 parallaxes with uncertainty
+less than 10%. The bin size is 0.25 mag (see text). Adapted
+from [54].
+from CoRoT, Kepler, K2, and now TESS, have demon-
+strated the relevance and potential of this novel tech-
+nique in understanding stellar physics. The detection
+of frequencies of p and g-modes allows us to infer both
+average and localised properties of different internal re-
+gions of the target stars. In particular, the internal rota-
+tion rate may be measured, and important phenomena,
+such as gravity waves and magnetic buoyancy, may be
+constrained. In addition, the chemical composition gra-
+dient, the thermal stratification and the sound speed
+profile could be also deduced from frequency patterns.
+The next ESA missions, PLATO and, hopefully, HAY-
+DIN, will deliver more accurate data to improve our
+understanding of these processes.
+Finally, we can easily guess that all these future
+studies, thanks to the superior quality of the observa-
+tional data, will motivate and boost the development of
+new and more sophisticated stellar models, capable to
+accurately describe the non-standard processes involved
+in the formation and evolution of intrinsic C star.
+Acknowledgements This work has been supported by the
+Spanish project PGC2018-095317-B-C21 financed by the MCIN
+/AEI FEDER “Una manera de hacer Europa”, and by the
+project PID2021-123110NB-I00 financed by MCIN/AEI
+/10.13039/501100011033/FEDER, UE.
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+arXiv:1310.3163 (2013)
+

6NAzT4oBgHgl3EQfvP1t/content/tmp_files/2301.01703v1.pdf.txt

@@ -0,0 +1,862 @@
+1
+Technology Trends for Massive MIMO towards 6G
+Yiming Huo, Senior Member, IEEE, Xingqin Lin, Senior Member, IEEE, Boya Di, Member, IEEE, Hongliang
+Zhang, Member, IEEE, Francisco Javier Lorca Hernando, Ahmet Serdar Tan, Shahid Mumtaz, Senior
+Member, IEEE, ¨Ozlem Tu˘gfe Demir, Member, IEEE, and Kun Chen-Hu, Member, IEEE
+Abstract—At the dawn of the next-generation wireless systems
+and networks, massive multiple-input multiple-output (MIMO)
+has been envisioned as one of the enabling technologies. With the
+continued success of being applied in the 5G and beyond, the mas-
+sive MIMO technology has demonstrated its advantageousness,
+integrability, and extendibility. Moreover, several evolutionary
+features and revolutionizing trends for massive MIMO have
+gradually emerged in recent years, which are expected to reshape
+the future 6G wireless systems and networks. Specifically, the
+functions and performance of future massive MIMO systems will
+be enabled and enhanced via combining other innovative tech-
+nologies, architectures, and strategies such as intelligent omni-
+surfaces (IOSs)/intelligent reflecting surfaces (IRSs), artificial
+intelligence (AI), THz communications, cell free architecture.
+Also, more diverse vertical applications based on massive MIMO
+will emerge and prosper, such as wireless localization and sens-
+ing, vehicular communications, non-terrestrial communications,
+remote sensing, inter-planetary communications.
+Index Terms—6G, Massive MIMO, Intelligent Omni-Surface
+(IOS), Intelligent Reflecting Surface (IRS), Cell Free, Artifi-
+cial Intelligence, Vehicular Communications, THz Communica-
+tions, Non-Terrestrial Communications, Remote Sensing, Inter-
+Planetary Communications.
+I. INTRODUCTION
+Massive multiple-input multiple-output (MIMO) has been
+one of the essential technologies in 5G wireless communi-
+cations and recently experiencing unprecedented growth in
+development and deployment. With fast technological inno-
+vations and enormous commercial needs, mass MIMO is
+expected to evolve further and reshape future telecommuni-
+cations and related areas more broadly and deeply.
+Support for massive MIMO is intrinsic in 5G New Radio
+(NR) standards [1]. As the first release of 5G NR, Release 15
+includes the fundamental features to support massive MIMO
+in different deployment scenarios, including reciprocity-based
+operation for time division duplex (TDD) systems, high-
+resolution channel state information (CSI) feedback for multi-
+Yiming Huo is with the Department of Electrical and Computer Engineer-
+ing, University of Victoria, Victoria, BC V8P 5C2, Canada (ymhuo@uvic.ca).
+Xingqin
+Lin
+is
+with
+NVIDIA,
+Santa
+Clara,
+CA
+95050,
+USA
+(xingqinl@nvidia.com).
+Boya Di, and Hongliang Zhang are with the School of Electronics,
+Peking University, Beijing 100871, China (e-mail: boya.di@pku.edu.cn;
+hongliang.zhang92@gmail.com).
+Francisco Javier Lorca Hernando, and Ahmet Serdar Tan are with Inter-
+Digital Communications, Inc. London, England, United Kingdom (e-mail:
+javier.lorcahernando@interdigital.com; AhmetSerdar.Tan@interdigital.com).
+Shahid Mumtaz is with the Instituto de Telecomunicac¸˜oes, Aveiro, Portugal
+(e-mail: smumtaz@av.it.pt).
+¨Ozlem Tu˘gfe Demir is with the Department of Computer Science, KTH
+Royal Institute of Technology, Stockholm, Sweden (e-mail: ozlemtd@kth.se).
+Kun Chen-Hu is with the Department of Signal Theory and Communica-
+tions of Universidad Carlos III de Madrid, Legan´es, 28911, Spain (e-mail:
+kchen@tsc.uc3m.es).
+user MIMO (MU-MIMO), and advanced beam management
+for high-frequency band operation with analog beamforming,
+among others. After Release 15, 3GPP specified further en-
+hancements of massive MIMO in Release 16. Representa-
+tive massive MIMO enhancements in Release 16 are CSI
+feedback overhead reduction through spatial and frequency
+domain compression, beam management signaling overhead
+and latency reduction, and non-coherent joint transmission
+from multiple transmit and receive points (TRPs).
+3GPP continued massive MIMO evolution in Release 17.
+CSI feedback overhead was further reduced by exploiting
+angle-delay reciprocity. A unified transmission configuration
+indicator (TCI) framework was introduced to enhance multi-
+beam operation. Multi-TRP support was also improved with
+the introduction of inter-cell multi-TRP enhancements and
+multi-TRP-specific beam management features. Release 18 is
+the start of work on 5G Advanced, and its scope includes
+further massive MIMO evolution. Potential directions under
+investigation in 3GPP are uplink MIMO enhancements (e.g.,
+the use of eight transmission antennas in the uplink and multi-
+panel uplink transmission), an extension of the unified TCI
+framework from single TRP to multi-TRP scenarios, a larger
+number of orthogonal demodulation reference signal (DMRS)
+ports for MU-MIMO, and CSI reporting enhancements for user
+equipment (UE) with medium and high velocities.
+With fast standardization and promising commercialization,
+massive MIMO becomes the critical underlying technology
+for 5G and beyond, and is expected to combine other en-
+abling technologies and expand to more new verticals. This
+article presents and analyzes several technology trends for
+massive MIMO evolving on the path to 6G. For example,
+one of the critical observations is the recent intense attention
+paid to intelligent surfaces [2] which hold great potential to
+enable energy/cost-efficient massive MIMO. Furthermore, the
+intelligent reflecting surface (IRS) enabled massive MIMO
+can facilitate joint communications, localization and sensing
+functions that extensively enable new use cases and strengthen
+the wireless system performances in 6G.
+This article’s first two sections are dedicated to IRS physical
+fundamentals for massive MIMO and IRS-enabled massive
+MIMO for localization and sensing, respectively. Immediately
+afterward, we provide a survey on ultra-massive MIMO at
+THz frequencies since adopting small wavelengths, and wide
+bandwidth brings unprecedented challenges in almost every
+aspect of the wireless system design. Then, we investigate the
+cell-free massive MIMO technology which improves spectral
+and energy efficiency. Next, the artificial intelligence (AI)
+for massive MIMO is surveyed and discussed, followed by
+a review and discussion of massive MIMO-OFDM for high-
+arXiv:2301.01703v1  [cs.IT]  4 Jan 2023
+
+2
+Incident signal
+Refracted 
+signal
+MU 1
+Z
+X
+Y
+IOS element m
+(m)
+(m)
+Source
+MU 2
+Reflected 
+signal
+1(m)
+(m)
+1
+User 1
+User 2
+Fig. 1. Transmission model of an intelligent surface element.
+speed applications. As the last vertical of future massive
+MIMO towards 6G, non-terrestrial networks (NTNs) com-
+munications has been one direction in standardization since
+2017 [3]. Thus, we present a detailed review, discussion, and
+analysis of current and futuristic non-terrestrial applications
+and architectures on top of massive MIMO before concluding
+this article.
+II. IOS/IRS PHYSICAL FUNDAMENTALS FOR MASSIVE
+MIMO
+Driven by the explosive growth in wireless data traffic,
+there is pressing need for innovative communication paradigms
+supporting high data rates in the future 6G. Massive MIMO
+has attracted heated attention exploiting the implicit random-
+ness of the wireless environment. However, traditional massive
+MIMO relies on the large-scale phase arrays, which induce
+high hardware cost and power consumption due to the energy-
+consuming phase shifters, especially when the number of
+antennas grows. This limits their scalability to support massive
+MIMO in practice.
+Recently, intelligent metasurface, as a new type of ultra-
+thin two-dimensional metamaterial inlaid with sub-wavelength
+scatters, has provided a novel technology to enable massive
+MIMO in a cost-efficient way. Capable of reflecting and/or
+refracting the incident signals simultaneously, the surface can
+actively shape uncontrollable wireless environments into a
+desirable form via flexible phase shift reconfiguration [4].
+Since such reconfiguration of each element is usually achieved
+via one or two PIN diodes controlled by the biased voltage,
+it only involves little hardware and power costs compared to
+the traditional phase arrays. Thus, the surface can be easily
+extended to a large scale, providing a practical method for
+realizing massive MIMO.
+A general transmission model of one surface element is
+shown in Fig.1. After the incident signal arrives at the surface,
+part of it is reflected and the rest is refracted towards the other
+side. By defining the reflection-refraction ratio as ϵ, we have
+the reflected and refracted signals in Fig.1. Three different
+types of surfaces can then be classified below:
+• When ϵ = 0, the surface only reflects the incident signal,
+leading to an intelligent reflecting surface. It can be
+attached to the wall, serving as a reflective relay for
+coverage.
+• When ϵ → ∞, the surface only refracts the incident
+signal, serving as an reconfigurable refractive surface
+(RRS). It can replace the antenna array at the base station
+for transmission and reception.
+• When 0 < ϵ < ∞, the surface can reflect and refract
+the incident signal simultaneously, named as intelligent
+omni-directional surface (IOS). Compared to IRS, it
+can achieve full-dimensional wireless communications
+despite users’s locations with respect to the surface.
+Both IOS and IRS have been considered as efficient methods
+to achieve massive MIMO due to their mature implementation.
+Especially, the recently developing IOS technique has also
+brought new challenges to the field:
+• The refracted and reflected signals of IOS are coupled
+with each other, determined simultaneously by the states
+of PIN diodes. Such a coupling effect makes it unknown
+whether IOS has the same impact on EM waves when
+the signal impinges on different sides of the IOS, i.e.,
+whether the channel reciprocity still holds for the IOS-
+aided transmissions.
+• Besides, to fully exploit the refract-and-reflect character-
+istic of IOS, it is also necessary to explore the optimal
+position and orientation of the IOS given the BS and user
+distribution to extend the coverage on both sides of the
+IOS.
+• In addition, a beamforming scheme should be reconsid-
+ered and tailored for the IOS-aided transmission since the
+reflected and refracted beams towards different users are
+dependent with each other [5].
+III. LOCALIZATION AND SENSING USING IOS/IRS
+ENABLED MASSIVE MIMO
+In future 6G, wireless localization and sensing functions
+will be built-in for various applications, e.g., navigation, trans-
+portation, and healthcare. As a result, it is highly demanding
+to provide services with fine-resolution sensing and high
+localization accuracy. To realize this vision, massive MIMO
+can be a promising solution as the beam width can be reduced
+with a larger antenna array, leading to a high spatial reso-
+lution. However, the wireless environments in these systems
+are becoming complicated, for example, line-of-sight (LoS)
+links might be blocked by buildings or objects, degrading
+the accuracy of sensing and localization. Fortunately, the
+development of the IOS can provide favorable propagation
+conditions to improve sensing and localization accuracy [6].
+On the one hand, the IOS can provide additional paths toward
+targets, extending the coverage. On the other hand, with the
+capability of manipulating propagation conditions, the signals
+from different objects or targets can be customized so that they
+are easier to be distinguished, as illustrated in Fig. 2.
+Nevertheless, the integration of the IOS in a wireless sens-
+ing/localization system is not trivial, which generally brings
+the following challenges:
+• It will be a challenge to optimize the configurations
+relating to the IOS. Different from the designs of the IOS
+
+3
+Tx
+Body Sensing
+Additional paths
+Rx
+Smartphone 
+positioning
+1
+2
+3
+4
+5
+Configurations
+Customized 
+signal strength
+Fig. 2. Illustration for wireless localization and sensing using IOS enabled
+massive MIMO systems.
+for the communication purposes, the optimizations here
+aim to maximize the sensing/localization performance,
+necessitating new designs. For example, the metric could
+be defined as the distance between two signal patterns
+(each corresponding to a configuration of the IOS) from
+different targets/positions so that the receiver could rec-
+ognize two targets/positions with less efforts, leading
+to a higher accuracy. Moreover, as the number of IOS
+elements could be large, it will cause prohibitively high
+delay to enumerate all the configurations. Therefore,
+it will be important to select an appropriate number
+of configurations to achieve the trade-off between the
+latency and accuracy.
+• The coupling of decision function with the optimization
+of the IOS makes it hard to find the optimal function.
+To be specific, the receiver needs a decision function
+to transform the received signals into the information of
+targets/positions. As the received signals can be adjusted
+by the IOS, the selection of decision function is also
+influenced by the configurations of the IOS. Therefore,
+a joint optimization will be necessary to improve the
+performance [6].
+• In addition to the above signal processing challenges,
+practical implementation is another challenge. Where to
+deploy the IOS and how to determine its size should be
+carefully addressed, which should also take the topology
+of the environment into consideration.
+To sum up, massive MIMO technology will be expected to
+provide multi-functional services integrating communication,
+localization, and sensing. The IOS, which could customize the
+propagation environments, is believed to be an add-on enabler
+for future massive MIMO to facilitate such an integration.
+IV. ULTRA-MASSIVE MIMO AT THZ FREQUENCIES
+According to ITU-R (International Telecommunication
+Union Radiocommunication Sector), THz frequencies are
+those in the range 0.1 THz – 10 THz. The lowest frequency
+region between 0.1 THz and 0.3 THz with the highest po-
+tential is usually called the sub-THz regime. THz and sub-
+THz signals serve as a bridge between radio and optical
+frequencies. Their wavelengths in the millimeter and sub-
+millimeter region make them excellent candidates to fulfill the
+6G promise of extremely high-capacity communications, good
+situational awareness, and ultra-high resolution environmental
+sensing. Such small wavelengths come at the price of high
+uncertainty in the channel characteristics, leading to unreliable,
+intermittent radio links that suffer from one or several of the
+following impairments:
+• High path losses, molecular absorption, and blockage.
+The high free-space path loss motivated by the small
+antenna aperture areas at these frequencies, together with
+the molecular absorption, blockage, diffuse scattering,
+and extra attenuation caused by rain, snow, or fog, lead to
+highly intermittent links. Link reliability must therefore
+be improved with the use of ultra-narrow beamforming.
+• Low energy efficiency. RF output power degrades 20 dB
+per decade for a given Power Amplifier (PA) technology.
+This compromises the link budget and reinforces the need
+of large-scale transceivers with high numbers of antennas.
+• Large-scale transceivers. The high beamforming gain
+needed to improve link reliability demands large-scale
+transceivers with a high number of antennas (usually,
+more than 1024). The sharpened, ultra-narrow beams that
+they produce pose significant challenges to mobility and
+beam tracking.
+• Phase noise. At sub-THz/THz frequencies, CP-OFDM
+performance can be severely degraded by the inter-carrier
+interference (ICI) resulting from phase noise. Increasing
+the subcarrier spacing can mitigate its impact, but the
+correspondingly shorter symbol duration introduces a
+penalty in coverage and impairs the ability to mitigate
+large delay spreads.
+• Channel sparsity. Ultra-narrow beams, together with ray-
+like wave propagation, lead to channels that exhibit small
+numbers of spatial degrees of freedom and ranks limited
+to one LoS component and a few multipath components,
+which challenges MIMO operation.
+• Spherical
+wave
+and
+near-field
+effects.
+Large-scale
+transceivers exhibit significant spherical wave and near-
+field effects from the electrically large antenna structures
+that they equip, which introduces complexity to MIMO
+precoding strategies.
+• Beam squint. The narrowband response of phase shifters
+in planar arrays introduces a frequency-dependent beam
+misalignment called beam squint. Losses from beam
+misalignments can be alleviated by using beam broad-
+ening techniques, at the cost of reduced coverage; and
+avoided with true time delay (TTD) units, at the cost of
+complexity.
+There is abundant research on transceiver architectures and
+network solutions aimed to ameliorate some of the above is-
+sues, especially those motivated by the propagation challenges
+at sub-THz/THz. Among the network solutions, the aforemen-
+tioned IRS/RIS equipped with very large numbers of small
+antenna elements are receiving considerable attention, because
+of their ability to tailor the characteristics of the reflected
+and refracted beams [2], [7]. IRS/RIS at sub-THz/THz exploit
+
+4
+ray deflections to overcome blocking and path loss; can take
+benefit of the near-field effects by focusing beams to improve
+beamforming and 3D imaging; and can enhance the multipath
+richness of the channel to reinforce the spatial multiplexing
+capabilities at sub-THz/THz frequencies.
+V. CELL-FREE MASSIVE MIMO
+Cell-free massive MIMO is envisioned as a promising
+technology for beyond 5G systems due to the highly improved
+spectral and energy efficiency it would provide. As a natural
+consequence, not only the academia but also the industry has a
+great interest in cell-free massive MIMO, which is also named
+“distributed MIMO” or “distributed massive MIMO” by in-
+dustrial researchers [8]. It aims to guarantee almost uniformly
+great service to every user equipment in the coverage area by
+benefiting from joint transmission/reception and densely de-
+ployed low-cost access points with increased macro diversity.
+The physical-layer aspects such as receiver combining design,
+transmit precoding design, and power allocation algorithms in
+line with a futuristic scalable system design have now been
+well-established. For a scalable (in terms of signal processing
+complexity and fronthaul signaling load) cell-free massive
+MIMO system, an access point can only serve a finite number
+of user equipments. One service-oriented design option is the
+user-centric formation of the access points serving each user
+equipment according to their needs. As illustrated in Fig. 3,
+each user equipment is served by multiple access points with
+the preferable channel conditions, which are the ones in the
+colored shaded circular regions.
+The centralized computational processing unit and the fron-
+thaul links between it and access points are two major layers
+in a practical cell-free massive MIMO operation envisioned
+to be built in 6G communication systems. When edge clouds
+are placed between the access points and the center cloud, as
+shown in Fig. 3, the midhaul transport and the collaborative
+processing unit consisting of the edge and center cloud are
+the additional components in a cell-free network. Hence, the
+imperfections, limitations, and energy consumption should be
+analyzed from an end-to-end (from radio edge to the center
+cloud) perspective. Conducting an end-to-end study of a low-
+cost and energy-efficient cell-free massive MIMO implemen-
+tation is critical to accelerating its practical deployment in 6G.
+The network architecture of a cell-free massive MIMO
+system with access points connecting to central processing
+units via fronthaul links is entirely in line with the wave of
+cloudification in mobile communications networks. Hence, it
+is expected from the very beginning to envision prospective
+cell-free networks on top of a cloud radio access network
+(C-RAN). Virtualized C-RAN enables centralizing the digital
+units of the access points in an edge or central cloud with vir-
+tualization and computing resource-sharing capabilities. Going
+beyond virtualized C-RAN, the implementation options of
+cell-free massive MIMO have been discussed on top of open
+radio access networks (O-RAN) aiming for an intelligent,
+virtualized, and fully interoperable 6G architecture [9].
+Fronthaul/midhaul transport technology is one of the vital
+components in the low-cost deployment of cell-free massive
+User Equipments
+Access Points
+Edge Cloud
+Center Cloud
+Midhaul 
+Fronthaul
+Fig. 3. The C-RAN architecture with cell-free massive MIMO functionality.
+MIMO onto the legacy network. In a large-scale cell-free mas-
+sive MIMO system, deploying a dedicated optical fiber link
+between each access point and the edge or central cloud would
+be highly costly and infeasible. The so-called “radio stripes”-
+based fronthaul architecture developed by Ericsson reduces the
+cabling cost by sequentially integrating the access points into
+the shared fronthaul lines. When access points are distributed
+in a large area, other low-cost fronthaul transport technologies
+such as millimeter wave and terahertz wireless can both
+provide huge bandwidth and avoid costly wired fiber links.
+One other option is combined fiber-wireless fronthaul/midhaul
+transport to balance a trade-off between link quality and cost.
+In the latter method, the short-distance fronthaul links can
+be deployed wirelessly between each access point and its
+respective edge cloud. On the other hand, the midhaul transport
+from the edge to the center can benefit from extra-reliable fiber
+connections. Mitigating hardware impairments that naturally
+appear as a result of low-cost transceivers deployed at the
+access points and wireless fronthaul nodes is another critical
+aspect of the cell-free massive MIMO deployment on the
+legacy network.
+In recent years, energy-saving techniques by mobile oper-
+ators have gained more importance in reducing the environ-
+mental footprint and designing next-generation mobile com-
+munication systems in a green and sustainable way. Several
+works considered access point switching on/off methods in
+this research direction to save energy in a cell-free massive
+MIMO system. In addition, the virtualization and sharing of
+cloud and fronthaul/midhaul resources are crucial for mini-
+mizing total end-to-end energy consumption. At the end of the
+day, one should consider the limitations, energy consumption
+models, and the energy-saving mechanisms of digital units
+and processors in the edge and center cloud for the complete
+treatment of energy efficiency in a cell-free massive MIMO
+system.
+VI. ARTIFICIAL INTELLIGENCE FOR MASSIVE MIMO
+Massive MIMO technology powered by artificial intelli-
+gence (AI) can be applied to Industry 5.0. This technology
+can realize highly reliable real-time transmission of industrial
+
+5
+6G and other information with reliable human-computer in-
+teraction (HCI). Massive MIMO is a core technology of 5G.
+Still, the increase in the number of antennas has brought new
+challenges, significantly the rapid growth in the cost of channel
+estimation and feedback. Moreover, the accuracy of channel
+estimation and prediction needs to be improved. The applica-
+tion of AI technology is expected to solve the above problems.
+However, the above up-and-coming technologies have some
+issues. On the one hand, from the industry perspective, the
+main challenges are:
+• It is not easy to effectively control the difference between
+the training data set and the actual channel. The lack of
+generalization of AI algorithms may lead to a decline in
+system performance.
+• Wireless AI data and applications have their unique
+characteristics. However, how to organically integrate the
+classic AI algorithms in image and voice processing with
+wireless data is still unclear.
+• One of the characteristics of the Massive MIMO com-
+munication system applied to Industry 5.0 is that the
+communication scenarios are complex and changeable
+(indoor, outdoor, etc.), and the business forms are diverse.
+Therefore, making the wireless AI solution applicable
+to various communication scenarios and business forms
+under limited computing power is a significant challenge
+that the industry needs to overcome.
+On the other hand, there are several interesting trends from
+the research perspective. First, applying machine learning into
+resource allocation has the potential to achieve low complexity
+implementation and decrease operational costs for massive
+MIMO. This strategy can improve spectral efficiency and
+energy efficiency, increase the number of users, and decrease
+energy consumption as well as the time delay. Second, using
+machine learning or deep learning for signal detection in mas-
+sive MIMO has the potential to mitigate the high complexity
+issues seen in the conventional linear and non-linear detection
+methods. Third, AI can play a potential role in interference
+management for massive MIMO, such as determining and
+predicting the number of interference sources and strengths,
+and further mitigating the interference. Last but not least,
+with Massive MIMO expanding to more verticals, developing
+and deploying suitable segmented AI strategies for specific
+applications is critical.
+VII. MASSIVE MIMO-OFDM FOR HIGH-SPEED
+APPLICATIONS
+For massive MIMO, a very large number of antennas is used
+to either to reduce the multi-user interference (MUI), when
+spatially multiplexing several users, or to compensate the path
+loss when higher frequencies than microwave are used, such
+as the millimeter-waves (mm-Waves). Usually, a coherent de-
+modulation scheme (CDS) is used in order to exploit MIMO-
+OFDM (orthogonal frequency-division multiplexing), where
+the channel estimation and the pre/post-equalization processes
+are complex and time-consuming operations, which require a
+considerable pilot overhead and increase the latency of the
+system. Moreover, new challenging scenarios are considered
+Fig. 4.
+Massive MIMO for high-speed applications, a design example of
+hybrid demodulation scheme.
+in 5G and beyond, such as high mobility scenarios (e.g.,
+vehicular communications). The performance of the traditional
+CDS is even worse since reference signals cannot effectively
+track the fast variations of the channel with an affordable
+overhead.
+As an alternative solution, non-coherent demodulation
+schemes (NCDS) based on differential modulation combined
+with massive MIMO-OFDM have been proposed [10]. It is
+shown that even in the absence of reference signals, they can
+significantly outperform the CDS with a reduced complexity in
+high-speed scenarios, where no reference signals are required.
+In order to successfully implement the NCDS with the MIMO-
+OFDM system, some relevant details should be noted as
+follows. First, the high number of antennas is a key aspect to
+successfully deploy the NCDS. In the uplink, these antennas
+are used as spatial combiner capable of reducing the noise
+and self-interference produced by the differential modulation.
+In the downlink, beamforming is combined with NCDS in
+order to increase the coverage and spatially multiplex the
+different users. Then, the differential modulation should be
+mapped in the two-dimensional time-frequency resource grid
+of the OFDM symbol. Different schemes are proposed: time
+domain, frequency domain and hybrid domain, where the
+latter exhibits the best performance since it can minimize
+required signaling to a single pilot symbol for each transmitted
+burst. Last, on top of the MIMO-OFDM system, multiple
+users can be multiplexed in the constellation domain, which
+is an additional dimension to the existing spatial, time and
+frequency dimensions. At the transmitter, each user is choosing
+its own individual constellation, while at the receiver, the
+received joint constellation is a superposition of all individual
+constellations of each user. The overall performance in terms
+of bit error rate (BER) depends on the design of the received
+joint constellation, all chosen individual constellation and the
+mapped bits of each symbol. This non-convex optimization
+problem is solved using evolutionary computation, which is a
+subfield of artificial intelligence, capable of solving this kind
+of mathematical problems.
+Finally, in those low or medium-mobility scenarios, a hy-
+brid demodulation scheme (HDS) is proposed in [11], which
+consists of replacing the traditional reference signals in CDS
+by a new differentially encoded data stream that can be non-
+coherently detected. The latter can be demodulated without
+the knowledge of the channel state information and subse-
+quently used for the channel estimation. An design example
+
+n=1n=2n=3n=4n=5
+n=6
+n=7n=8n=9n=10n=11n=12n=13n=14
+k=1
+P
+P
+k 2
+high
+k=3
+k=4
+k=5
+k 6
+NCDS
+k=7
+k=8
+k=9
+k=10
+Hybrid:
+k=11
+k=12
+Dopplerspread
+CDS+NCDS
+coherent data
+non-coherent data
+CDS
+MOI
+(PSAMorST)
+low
+high
+Delay spread (1≤ K, ≤ 12)6
+is illustrated in Fig. 4. Consequently, HDS can exploit both
+the benefits of a CDS and NCDS to increase the spectral
+efficiency. It is outlined that the channel estimation is almost
+as good as CDS, while the BER performance and throughput
+are improved for different channel conditions with a very small
+complexity increase.
+VIII. MASSIVE MIMO FOR NON-TERRESTRIAL
+COMMUNICATIONS
+With 5G standardization phase 1 & 2 finalized in 3GPP
+release 15 & 16, the first half of 2022 has witnessed the
+third installment of the global 5G standard reaching the system
+design completion in 3GPP release 17 deemed as a continued
+expansion to 5G new devices and applications. In particular,
+3GPP release 17 has introduced the 5G NR support for satellite
+communications which is one critical family member of the
+non-terrestrial networks (NTN). In fact, the concept of the
+NTN encompasses any network involving flying objects in
+either the air or space, and the NTN family therein includes
+satellite communication networks, high-altitude platform sys-
+tems (HAPS) (including airplanes, balloons, and airships), and
+air-to-ground networks [3].
+As the focus of 3GPP NTN work, the satellite communi-
+cation networks enable advanced features such as ubiquitous
+connectivity and coverage for remote/rural areas. Moreover,
+they include two distinct aspects, one focusing on satellite
+backhaul communications for application scenarios such as
+customer-premises equipment (CPE) and direct low data rate
+services for handhelds, while another one aims at adapting
+eMTC (enhanced Machine Type Communication) and NB-
+IoT (Narrowband Internet-of-Things) operation to satellite
+communications. Recent years have witnessed the unprece-
+dented interest and prosperity in the Low Earth orbit (LEO)
+satellites enabled broadband access and services [12]. Among
+several major commercial players in the arena, namely Starlink
+(SpaceX), Kuiper (Amazon), OneWeb, Boeing, and Telestat,
+Starlink leads in terms of scale and dimension of satellite
+megaconstellation and number of service subscribers.
+There are several essential catalysts for accelerating the fast-
+booming spaceborne broadband access [12], such as the launch
+cost decrease, private capitals, wide deployment of AI and
+cloud/edge computing, and high-performance satellite wireless
+and networking technologies. From wireless communications
+and a particular massive MIMO perspective, an overview of
+trends and challenges is presented as follows.
+• Deploying and operating expanding satellite constella-
+tions below 2,000 km brings potential challenges of han-
+dling competition and facilitating coexistence. In order to
+minimize the propagation delay, many LEO megaconstel-
+lation builders may want to deploy satellites in orbits as
+low as possible, while the ITU adapts the “First Come,
+First Served” approach in the ITU cooperative system to
+access orbit/spectrum resources. With the space in the
+LEO becoming more crowded and collision incidents be-
+ing reported [12], regulations, strategies, and technologies
+are required to cope with increasing space traffic and han-
+dle safe deorbiting/disposal of satellites/spacecraft. With
+Fig. 5. Illustration of massive MIMO for non-terrestrial networks (NTNs).
+the escape velocity being at 7.8 km/s, tracking, localizing
+the satellites/spacecraft and further enabling collision-
+avoidance can be difficult even with contemporary AI-
+assisted sensing and detecting technologies.
+• Moreover, spectrum management is another critical as-
+pect since the generally limited spectrum resource poses
+severe challenges. To facilitate 5G NR for NTN, 3GPP
+release 17 has investigated supporting satellites backhaul
+communication for CPEs and direct link to handhelds for
+low data rate services using sub-7 GHz S-band, while
+using frequency higher than 10 GHz will be studied in
+3GPP release 18. In the meantime, the first-generation
+system of Starlink, or Gen1, has mainly used Ku-band
+and Ka-band for different types of links and transmission
+directions, and its second-generation (Gen2) will add
+V-band into it. Either sub-7 GHz S-band or Ku/Ka/V-
+band, to some extent, will overlap with some spectrum
+of ongoing 5G and future 6G systems, and/or other
+systems operating in these bands. Consequently, there
+can be interference and co-existence challenges among
+different systems and networks. In fact, both SpaceX
+and OneWeb have expressed concerns about the possible
+interference experienced by the non-geostationary orbit
+(NGSO) satellite internet if the terrestrial 5G uses 12
+GHz band. Furthermore, supporting more satellite direct
+links to the user equipment (UE) using sub-7 GHz S-band
+also makes this interference challenge more pronounced.
+More studies of spectral resources (e.g. higher frequency
+bands) and spectrum management for spaceborne massive
+MIMO are expected.
+• Furthermore, there are various types of interferences that
+could emerge both within the same space network and
+among different space networks. For example, the in-
+band/out-band interference (or emission) can happen for
+user terminals (UTs) and ground stations of the same
+megaconstellation. When it comes to the situation of mul-
+tiple space networks, satellite transmission of one mega-
+
+Mars
+Moon
+Satellite
+networks
+HAPS
+Aerial
+Maritime
+Remote area
+Suburban
+Urban7
+constellation could cause interference to the reception by
+UTs and ground stations belonging to other megaconstel-
+lations. Also, the UTs/ground stations transmission could
+interfere with the satellite(s) in different constellations.
+Conventionally, co-existing space networks need to share
+frequency allocations (both uplink and downlink) with
+each other to mitigate the interference, which is based on
+the coordination. However, more high-performance inter-
+ference mitigation technologies are required for coping
+with more complicated situations in the future.
+• From a big-picture viewpoint, one the one hand, there
+is a trend that NGSO megaconstellations will support
+or efficiently co-work with GEO (geosynchronous Earth
+orbit) networks, HAPS, air-to-ground networks, drone
+networks, etc. Enabling such a greater NTN eco-system
+can bring more challenges of handling co-existence and
+competition. On the other hand, the space-enabled net-
+works and massive MIMO will be expanded to and
+beyond the near-Earth space (NES) which is the space
+from the layers of the neutral terrestrial atmosphere (160-
+200 km) up to the lunar orbit (around 384,400 km). For
+example, NOKIA Bell Labs will deploy the first LTE
+network on the Moon for NASA’s Artemis program. In
+the proposed Solar Communication and Defense Net-
+works (SCADN) concept [13], a massive MIMO sensing
+and communications framework based on an internet of
+a large number of spacecraft/satellites across the entire
+solar system enables early detection and mitigation of
+potential threats (e.g. asteroid/comet) to Earth and extra-
+terrestrial human bases, and also provides infrastructure
+facility to wireless connectivity within the solar system
+before/when human presence establishes on other celes-
+tial bodies. The very large propagation distances (between
+Earth and another celestial bodies) and delays pose severe
+challenges to the wireless sensing and communications,
+which requires more innovative solutions such as artifi-
+cial intelligence, machine/deep learning, edge computing,
+edge AI, distributed and federated learning, etc.
+To sum up, the massive MIMO technology has been
+fast extending the communicating and sensing capabilities
+of humanity beyond the terrain and even Earth, which will
+undoubtedly facilitate a more prosperous space era for all
+mankind.
+IX. CONCLUSIONS
+This article presents a comprehensive overview of promising
+technology trends for massive MIMO on the evolving path
+to 6G. First, we conduct an overview of massive MIMO’s
+recent standardization and research progress. Then we focus
+on IRS/IOS technologies that can enable/cost-efficient mas-
+sive MIMO communications. Furthermore, we envision the
+challenges of using IRS/IOS for facilitating and enhancing
+the localization and sensing capabilities in massive MIMO.
+Next, we investigate the ultra-massive MIMO at THz frequen-
+cies and unveil several impairments that affect the system
+design. Then we present and analyze the cell-free massive
+MIMO architecture which can boost the spectral and energy
+efficiency of wireless systems and networks. In addition, the
+challenges and trends of AI for massive MIMO are discussed
+in depth. Meanwhile, future massive MIMO will enable and
+strengthen more critical vertical applications. Therefore, the
+massive MIMO-OFDM-enabled high-speed communications
+is surveyed and presented with some designed examples.
+Finally, we carefully present and analyze the current and
+future trends of massive MIMO communications for non-
+terrestrial networks, particularly near-Earth space and inter-
+planetary applications.
+ACKNOWLEDGMENTS
+We express sincere thanks to the IEEE Future Net-
+works Massive MIMO Working Group, and the Organizing
+Committee of the IEEE Future Networks Second Massive
+MIMO Workshop. All co-authors contributed equally in this
+manuscript.
+REFERENCES
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+

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+Topological stripe state in an extended Fermi-Hubbard model
+Sergi Juli`a-Farr´e,1, ∗ Lorenzo Cardarelli,2, 3 Maciej Lewenstein,1, 4 Markus M¨uller,2, 3 and Alexandre Dauphin1, †
+1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,
+Av.
+Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
+2Peter Gr¨unberg Institute, Theoretical Nanoelectronics,
+Forschungszentrum J¨ulich, D-52428 J¨ulich, Germany
+3Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
+4ICREA, Pg.
+Llu´ıs Companys 23, 08010 Barcelona, Spain
+Interaction-induced topological systems have attracted a growing interest for their exotic prop-
+erties going beyond the single-particle picture of topological insulators. In particular, the interplay
+between strong correlations and finite doping can give rise to nonhomogeneous solutions that break
+the translational symmetry. In this work, we report the appearance of a topological stripe state in
+an interaction-induced Chern insulator around half-filling. In contrast to similar stripe phases in
+nontopological systems, here we observe the appearance of chiral edge states on top of the domain
+wall. Furthermore, we characterize their topological nature by analyzing the quantized transferred
+charge of the domains in a pumping scheme. Finally, we focus on aspects relevant to observing such
+phases in state-of-the-art quantum simulators of ultracold atoms in optical lattices. In particular,
+we propose an adiabatic state preparation protocol and a detection scheme of the topology of the
+system in real space.
+Introduction.—In the last decade, the quest for ma-
+terials exhibiting intrinsic topological phases in the ab-
+sence of external fields has been the focus of very in-
+tense research [1–3]. The interaction-induced quantum
+anomalous Hall (QAH) phase [2, 4–8], or chiral spin
+liquids [3, 9, 10], are two paradigmatic examples.
+In
+both cases, the interplay between the interactions and
+the geometry leads to the spontaneous breaking of time-
+reversal symmetry, and the resulting phases possess non-
+trivial topological invariants.
+The theoretical search
+of such interaction-induced topological phases in many-
+body systems has been further boosted by the devel-
+opment of tensor network approaches [11, 12]. In par-
+ticular, state-of-the-art density matrix renormalization
+group (DMRG) studies [13] in cylinder geometries have
+unambiguously established the presence of spontaneous
+Chern insulators in the ground-state phase diagram of
+several two-dimensional lattice models.
+These include
+effective models of twisted bilayer graphene [14], or ex-
+tended Fermi-Hubbard models of spinless fermions [15–
+17] that can be engineered in cold atom quantum simula-
+tors. Furthermore, fractional Chern insulators have been
+also identified in the spinful Fermi-Hubbard model [18]
+and in the Heisenberg model [19], both representing cases
+in which the system realizes a chiral spin liquid phase.
+While all these studies focused on spatially homoge-
+neous phases at commensurate particle fillings, it is worth
+noticing that the study of inhomogeneous phases at in-
+commensurate fillings, i.e., at finite doping, is of particu-
+lar interest. Several works in this direction have pushed
+tensor network simulations to their limit in order to iden-
+tify antiferromagnetic stripe domain walls of high-Tc su-
+perconductors in the underdoped region of the Hubbard
+model [20, 21], as first predicted by mean-field stud-
+ies [22, 23]. In the case of interaction-induced Chern insu-
+lating phases, very recent mean-field studies [24–27] sug-
+gested that at incommensurate dopings these systems can
+also exhibit domain walls between phases characterized
+by different topological invariants, leading to interaction-
+induced chiral edge states [24]. Remarkably, this picture
+is consistent with the subsequent experimental observa-
+tion of a mosaic of patches with opposite topological in-
+variants in twisted bilayer graphene [28].
+In this work, we analyze the phenomenon of spatially
+inhomogeneous topological phases in 2D. Based on a
+DMRG study in the matrix-product-state (MPS) rep-
+resentation, we confirm the numerical stability of these
+phases beyond the mean-field approximation in a cylin-
+der geometry with a very long length and short trans-
+verse direction.
+We also introduce techniques to mea-
+sure topological invariants in inhomogeneous systems and
+in a purely many-body scenario, i.e., beyond the single-
+particle approximation.
+To this aim, we consider the effect of doping in
+the interaction-induced homogeneous QAH phase of a
+fermionic lattice model. We start by showing that such a
+system indeed exhibits a topological stripe state, hosting
+chiral edge states at the domain walls. We then char-
+acterize the topological nature of the domains by means
+of a topological pumping scheme. Following Laughlin’s
+Gedankenexperiment [29], which we generalize to the in-
+homogeneous case, we extract the Chern number of the
+domains from their quantized charge transfer under an
+adiabatic flux insertion in the DMRG simulations.
+Our study not only reveals the fundamental features
+of these inhomogeneous solutions, and how they can be
+characterized in a strongly-correlated scenario. It is also
+further motivated by the prospect of quantum simulating
+these phases with cold atoms in optical lattices. In this
+regard, notice that in solid-state materials the QAH has
+arXiv:2301.03312v1  [cond-mat.quant-gas]  9 Jan 2023
+
+2
+only been observed in a few systems with spin-orbit cou-
+pling [4–6] or with interacting magnetic orbitals [7, 8].
+On the other hand, noninteracting Chern insulators have
+also been observed in quantum simulators [30–33] via
+the engineering of artificial gauge fields [34, 35].
+The
+extension of these experiments to the interacting case
+would allow one to observe new phenomena. Motivated
+by these reasons, we propose schemes to prepare these
+phases in an experiment and develop strategies to char-
+acterize their topology in real space.
+In this context,
+we show that the topological phase of the model could
+be prepared in a quasi-adiabatic protocol via a control
+parameter of the lattice that induces a continuous topo-
+logical phase transition. Such a result is essential to pro-
+vide a path to adiabatically prepare the phase in an ex-
+perimental setup.
+We finally discuss the possibility of
+measuring the topological nature of the phase through
+snapshot measurements of the particle density.
+Model.—We consider the extended Fermi-Hubbard
+Hamiltonian of spinless fermions on a checkerboard lat-
+tice described by the Hamiltonian ˆH = ˆH0 + ˆHint. The
+quadratic part ˆH0 of the Hamiltonian reads
+ˆH0 = − t
+�
+⟨ij⟩
+(ˆc†
+i ˆcj + H.c.) + J
+�
+⟨⟨ij⟩⟩
+eiφij(ˆc†
+i ˆcj + H.c.),
+(1)
+where t and J are the nearest-neighbor (NN) and next-
+to-nearest-neighbor (NNN) hopping amplitudes, respec-
+tively [see Fig. 1(a)]. The phase φij = ±π of the NNN
+tunneling generates a π-flux on each sublattice. On the
+other hand, the interacting part ˆHint of the Hamilto-
+nian has repulsive density-density interaction up to third
+neighbors and reads
+ˆHint = V1
+�
+⟨ij⟩
+ˆn′
+iˆn′
+j + V2
+�
+⟨⟨ij⟩⟩
+ˆn′
+iˆn′
+j + V3
+�
+⟨⟨⟨ij⟩⟩⟩
+ˆn′
+iˆn′
+j,
+(2)
+with ˆn′
+i ≡ ˆni − 1/2 and ˆni = ˆc†
+i ˆci.
+At half filling,
+ˆH0 exhibits two bands with a quadratic band touch-
+ing (semi-metallic phase). For finite interactions V1/2 ≃
+V2 ≫ V3, the frustration induced by the competition be-
+tween semi-classical charge orders allows for the emer-
+gence of an interaction-induced QAH state in the phase
+diagram [16, 24, 36]. The latter is characterized by the
+appearance of spatially homogeneous local current loop
+order, ξQAH ≡ �
+ij∈plaq. Im
+�
+ˆc†
+i ˆcj
+�
+, in NN plaquettes
+(see [37] for details), which breaks time-reversal sym-
+metry spontaneously. In addition, it is also character-
+ized by a nonzero value of a global topological invariant,
+the many-body Chern number ν. Importantly, there is
+an exact twofold ground state degeneracy, corresponding
+to the two opposite values of ξQAH. These two sectors
+are therefore characterized by opposite Chern numbers
+ν± = ±1.
+Topological stripe state.—We now discuss the appear-
+ance of spatially inhomogeneous Chern insulators in the
+FIG. 1.
+(a) Hopping processes of the Hamiltonian on the
+checkerboard lattice. (b) Sketch of the topological stripe state
+in the cylinder geometry. (c)-(e) Expectation value of local
+quantities integrated over the radial direction of the cylinder.
+(c) Current loop order featuring a sign inversion at the center
+of the cylinder. (d) Deviation of the local density from half
+filling. One can clearly observe the presence of the hole at
+the center. (e) Radial currents signaling the presence of chiral
+edge states in the regions where there is a change in the Chern
+number.
+model around half-filling, which constitutes one of the
+central results of this work. We consider a cylinder ge-
+ometry with 6 two-site unit cells in the radial direction
+(y) and 64 in the longitudinal one (x).
+To determine
+its ground state, we use the DMRG algorithm on the
+one-dimensional folding of the cylinder. In the numer-
+ical treatment, this 1D system, therefore, has effective
+long-range Hamiltonian terms, and one needs to use large
+bond dimensions χmax = 3000 in order to get trunca-
+tion errors of the order 10−5 at most.
+At half filling,
+for V1/t = 4.5, V2/t = 2.25, V3/t = 0.5 and J/t = 0.5,
+the system presents a degenerate QAH ground state with
+Chern numbers ν± = ±1. The addition of a single hole
+favors the breaking of the translational symmetry, as
+shown in Fig. 1. Figure 1(b) depicts the DMRG solution,
+which we call the topological stripe state. Such a state
+is spatially composed of two different Chern insulators,
+located on distinct halves of the cylinder and separated
+by a stripe domain wall. That is, due to the spontaneous
+breaking of translational invariance induced by doping,
+the two degenerate ground states of half-filling coexist
+in two separate regions of the same bulk. Figure 1(d)
+shows the density profile integrated along the radial di-
+rection.
+We observe that the domain wall is induced
+
+3
+FIG. 2. Quantized charge transport in the topological pump
+procedure performed with an adiabatic DMRG simulation.
+The net charges of the left, central, and right regions are
+shown in yellow, green, and blue colors, respectively, and as a
+function of the inserted flux θ, as indicated in the inset sketch.
+by the presence of a hole-like stripe, which is located in
+the bulk of the cylinder and has an integrated quantized
+charge of Q = −1 [38].
+The latter separates two dif-
+ferent Chern insulators, as signaled by the inversion of
+the current loop order ξQAH, shown in Fig. 1(c).
+No-
+tice that this is reminiscent of the change in the phase
+of the antiferromagnetic order parameter observed in the
+stripe phase of the Fermi-Hubbard mode, in the context
+of cuprate high-Tc superconductors [20, 21]. Here, how-
+ever, the local order parameter ξQAH is intertwined with
+the topological Chern number ν. This enriches the fea-
+tures of this topological stripe state, compared to the
+case of nontopological magnetic stripes.
+For instance,
+by virtue of the bulk-edge correspondence of topologi-
+cal insulators, one expects the presence of chiral edge
+states at the interface between the two different Chern
+insulators. Furthermore, these chiral edge states should
+have chiral current in the radial direction, defined as
+ξij
+y ≡ 2Jeiφij⟨ˆc†
+i ˆcj⟩, where (i, j) are NNN bonds in the
+radial direction. This quantity integrated in the radial
+direction is shown in Fig. 1(e). We observe positive net
+currents around the position of the hole, where the topo-
+logical invariant changes its value, as discussed below.
+Topological pump in inhomogeneous Chern insula-
+tors.—While the local quantities shown in Figs. 1(b)-(e)
+are consistent with a topological stripe state, where each
+of the sides of the cylinder has a different Chern num-
+ber, one needs to explicitly compute these global invari-
+ants in order to rigorously characterize the topological
+nature of this state. Notice that, for such an interact-
+ing and inhomogeneous state, this task is particularly
+challenging, as the main tools to study topology in real
+space, e.g., the local Chern marker [39, 40], are limited
+to the free fermionic picture, where interactions can only
+be treated in mean-field approximation [24, 41]. Here, to
+compute a spatially inhomogeneous Chern number in a
+purely many-body scenario, we follow the adiabatic flux
+insertion procedure, introduced as a gedankenexperiment
+by Laughlin [29]. This is based on the fact that Chern
+insulators exhibit a quantized Hall response equal to the
+Chern number after one cycle of a charge pump. While
+this method has been widely used in adiabatic DMRG
+simulations of homogeneous systems to compute their in-
+teger [15, 16] and fractional [18, 19] Chern numbers, here
+we show that it is also suited to analyze the topology
+of inhomogeneous stripe states. We insert a U(1) flux
+to the stripe ground state obtained in the previous sec-
+tion by adiabatically changing the phase of the tunneling
+terms crossing the y periodic boundary ˆc†
+i ˆcj → ˆc†
+i ˆcjeiθ in
+the DMRG simulation [42]. For a full cycle θ : 0 → 2π,
+and according to Laughlin’s argument, a homogeneous
+Chern insulator in a cylinder geometry pumps a quan-
+tized charge ∆Q equal to the value of |ν| from left to
+right, or vice versa, depending on the sign of the Chern
+number.
+For an inhomogeneous system with two dif-
+ferent nontrivial Chern numbers, we instead expect a
+quantized transport from the edges to the center, or vice
+versa, as discussed below. The effect of the flux inser-
+tion in the topological stripe state can be seen in Fig. 2,
+which shows the evolution of the integrated charge de-
+viation from half-filling, defined as QS,θ ≡ �
+i∈S ˆn′
+i(θ),
+where S ∈ {l, c, r} corresponds to the left, center, or
+right region of the cylinder, respectively. We also define
+the transferred charge on each region during the pump
+as ∆QS ≡ QS,2π − QS,0. At the beginning of the pump,
+Ql,0 = Qr,0 = 0, and Qc,0 = −1, as the added hole is
+located in the central region. As θ increases, the combi-
+nation of the Hall responses on each half of the cylinder
+leads to a net accumulation of charge in the domain wall,
+which indicates that the two halves of the cylinder have
+different Chern numbers. That is, for a unique value of
+the Chern number the charge would instead flow from
+one edge to the other without a net accumulation in the
+bulk. Indeed, notice that the charge pumped to the cen-
+ter domain wall is related to the Chern numbers of the
+left and right halves of the cylinder through
+∆Qc ≡ −(∆Ql + ∆Qr) = νl − νr.
+(3)
+At the end of the cycle (θ = 2π), we observe that both
+the left and right halves have transported a unit charge
+to the center, and the initial central hole is converted
+into a particle, i.e., ∆Qc = 2. This is in agreement with
+these two regions having different Chern numbers νl = 1
+and νr = −1. Therefore, with the help of Eq. (3) and the
+DMRG adiabatic flux insertion, we are able to unambigu-
+ously establish the topological character of this spatially
+inhomogeneous phase. For completeness, we also provide
+a qualitative single-particle explanation of this general-
+ized Laughlin pump for inhomogeneous systems in the
+
+4
+FIG. 3.
+Adiabatic state preparation of the interaction-
+induced QAH phase via the lattice control parameter M. The
+continuous behavior of the local order parameter ξQAH in-
+dicates a continuous phase transition from the trivial stripe
+insulator at M/t ≫ 1 to the QAH state at M → 0.
+Supplemental Materials [37].
+Adiabatic state preparation of the interaction-induced
+QAH phase.—Compared to other QAH states emerging
+from spontaneous symmetry breaking in solid-state sys-
+tems, the one considered here is described by a relatively
+simple Hamiltonian ˆH that can be quantum-simulated in
+a controlled environment. In particular, Rydberg-dressed
+atoms in optical lattices can be used to simulate such an
+extended Fermi-Hubbard model with tunable long-range
+interactions [36, 43].
+Here we focus on the yet unad-
+dressed question of the quantum state preparation of this
+exotic phase, which is ultimately related to the appear-
+ance of the domain wall states discussed above. For the
+adiabatic state preparation of the QAH phase [44, 45] it
+is desirable to find a second-order phase transition from
+a trivial insulator that could be easily initialized [46, 47].
+This strategy has already been used to prepare noninter-
+acting Chern insulators in optical lattices [48], and in the
+presence of interactions, there are numerical proposals
+to prepare fractional Chern insulators [49, 50]. The main
+difference in the present case is that the QAH phase arises
+from the spontaneous breaking of time-reversal symme-
+try in the ground state, that is, in the absence of exter-
+nal gauge fields. Therefore, we expect the appearance of
+Kibble-Zurek defects in a continuous transition [51–54],
+qualitatively resembling the static stripe state discussed
+above, and their interplay with topological chiral edge
+states.
+For the Hamiltonian ˆH under consideration, however,
+all the interaction-induced charge orders in the phase di-
+agram feature a first-order phase transition to the QAH
+state [16]. To overcome this problem, we propose to add
+to ˆH a staggering potential [50, 55] with strength M of
+−0.50
+−0.25
+0.00
+0.25
+⟨n′⟩
+(a)
+θ = 0
+0
+25
+50
+75
+100
+125
+x
+−0.50
+−0.25
+0.00
+0.25
+⟨n′⟩
+(b)
+θ = 2π
+FIG. 4. Computation of quantized Hall responses via local
+density snapshots in the topological pump procedure. (a),(b)
+Estimated density profiles from 3500 snapshots at the begin-
+ning and at the end of the flux insertion cycle, respectively.
+The Chern numbers of the left and right regions are extracted
+from the difference between these two cases.
+the form
+ˆHprep = M
+2
+�
+i
+(−1)si ˆni,
+(4)
+where si
+= ±1 on alternating two-site longitudinal
+stripes (see Fig. 3), which in the absence of interactions
+induces a local charge order at half filling corresponding
+to alternating empty and occupied stripes. In order to
+analyze the nature of the phase transition when varying
+M, we use the infinite density-matrix-renormalization-
+group (iDMRG) in the cylinder geometry with a single
+ring unit cell. Compared to the previous finite DMRG
+simulation of the topological stripe state, here we need to
+enlarge the bond dimension to χmax = 4000 to stabilize
+solutions with a small but finite value of the current loop
+order ξQAH.
+As shown in Fig. 3, when M dominates,
+the system is in a trivial charge insulating state with a
+vanishing current loop order ξQAH. Upon decreasing M,
+the local order parameter ξQAH becomes finite without
+exhibiting a clear discontinuous jump, which suggests a
+continuous phase transition to the QAH phase.
+Snapshot-based detection of the Chern number in
+transport experiments with cold atoms.—One of the ad-
+vantages of the numerical determination of Chern num-
+bers via the topological pump procedure described above
+is that it can be connected to the experimental measure-
+ment of this global topological invariant in real space.
+For instance, the 2D Laughlin topological pump itself has
+
+5
+been experimentally realized for noninteracting particles
+with cold atom quantum simulators in a synthetic cylin-
+der geometry [56]. Moreover, in a 2D lattice with open
+boundary conditions, the presence of an external force
+playing the role of an electric field is expected to result
+in the same quantized Hall response [57]. In both cases,
+the Chern number can be related to the charge drift in
+the system, which can be extracted from snapshots of
+the local density, accessible with a quantum gas micro-
+scope [58, 59]. Here, to numerically simulate snapshot
+measurements at the initial and final stages of the topo-
+logical pump, we use an algorithm proposed by Ferris
+and Vidal [60]. In a nutshell, this method allows one to
+efficiently draw independent snapshots of the local den-
+sity of an MPS, by simulating collapse measurements in
+the occupation basis at each site. The results are shown
+in Fig. 4, which shows the averaged values ⟨¯n′⟩ for 3500
+snapshots. In Fig. 4(a), corresponding to the θ = 0 case,
+the central hole is signaled by the depletion of the local
+density in this region. In this case, the deviation charges
+on the left and right regions are estimated, respectively,
+as Ql,0 = (0.01±0.26) and Qr,0 = (−0.01±0.26). At the
+final stage of the pump [Fig. 4(b)], one observes an excess
+charge in the central region, and the left and right regions
+have nonvanishing net charges of Ql,2π = (−0.95 ± 0.26)
+and Qr,2π = (−0.97 ± 0.26), respectively.
+From these
+quantities, we estimate the Chern number of the left and
+right regions as νl = (0.96±0.37) and νr = −(0.96±0.37),
+which are compatible with the ones extracted from Fig. 2.
+Conclusions.—We provided numerical evidence of a
+topological stripe state in an extended Fermi-Hubbard
+model at finite hole doping in a cylinder geometry. We
+generalized the numerical Laughlin pump procedure to
+characterize the two spatially separated Chern numbers
+of such a state. We furthermore discussed a related detec-
+tion scheme on a quantum simulator based on snapshot
+measurements of the local density. Our methods can be
+easily adapted to analyze other interacting systems with
+inhomogeneous topological properties in real space.
+Acknowledgments.—ICFO
+group
+acknowledges
+support
+from:
+ERC
+AdG
+NOQIA;
+Ministerio
+de
+Ciencia y Innovation Agencia Estatal de Investiga-
+ciones (PGC2018-097027-B-I00/10.13039/501100011033,
+CEX2019-000910-S/10.13039/501100011033,
+Plan Na-
+tional FIDEUA PID2019-106901GB-I00, FPI (reference
+code BES-2017-082118), QUANTERA MAQS PCI2019-
+111828-2, QUANTERA DYNAMITE PCI2022-132919,
+Proyectos de I+D+I “Retos Colaboraci´on” QUSPIN
+RTC2019-007196-7); MCIN Recovery, Transformation
+and
+Resilience
+Plan
+with
+funding
+from
+European
+Union NextGenerationEU (PRTR C17.I1);
+Fundaci´o
+Cellex; Fundaci´o Mir-Puig; Generalitat de Catalunya
+(European Social Fund FEDER and CERCA program
+(AGAUR Grant No.
+2017 SGR 134, QuantumCAT
+U16-011424, co-funded by ERDF Operational Program
+of Catalonia 2014-2020);
+Barcelona Supercomputing
+Center MareNostrum (FI-2022-1-0042);
+EU Horizon
+2020
+FET-OPEN
+OPTOlogic
+(Grant
+No
+899794);
+ICFO Internal “QuantumGaudi” project; EU Horizon
+Europe
+Program
+(Grant
+Agreement
+101080086
+—
+NeQST), National Science Centre, Poland (Symfonia
+Grant No. 2016/20/W/ST4/00314); European Union’s
+Horizon 2020 research and innovation program under the
+Marie-Sk�lodowska-Curie grant agreement No 101029393
+(STREDCH) and No 847648 (“La Caixa” Junior Lead-
+ers fellowships ID100010434: LCF/BQ/PI19/11690013,
+LCF/BQ/PI20/11760031,
+LCF/BQ/PR20/11770012,
+LCF/BQ/PR21/11840013).
+Views and opinions ex-
+pressed in this work are, however, those of the authors
+only and do not necessarily reflect those of the Eu-
+ropean Union, European Climate, Infrastructure and
+Environment
+Executive
+Agency
+(CINEA),
+nor
+any
+other granting authority. Neither the European Union
+nor any granting authority can be held responsible
+for them.
+The RWTH and FZJ group acknowledges
+support by the ERC Starting Grant QNets Grant
+Number
+804247,
+the
+EU
+H2020-FETFLAG-2018-03
+under Grant Agreement number 820495, by the Ger-
+many ministry of science and education (BMBF) via
+the VDI within the project IQuAn, by the Deutsche
+Forschungsgemeinschaft through Grant No. 449905436,
+and by US A.R.O. through Grant No.
+W911NF-21-
+1-0007, and by the Office of the Director of National
+Intelligence (ODNI), Intelligence Advanced Research
+Projects Activity (IARPA), via US ARO Grant number
+W911NF-16-1-0070. All statements of fact, opinions, or
+conclusions contained herein are those of the authors
+and should not be construed as representing the official
+views or policies of ODNI, the IARPA, or the US
+Government.
+The authors gratefully acknowledge the
+computing time provided to them at the NHR Center
+NHR4CES at RWTH Aachen University (project num-
+ber p0020074). This is funded by the Federal Ministry
+of Education and Research, and the state governments
+participating on the basis of the resolutions of the GWK
+for national high-performance computing at universities
+(www.nhr-verein.de/unsere-partner). The finite DMRG
+and iDMRG) simulations were performed using the
+iTensor [61] and TenPy [62] libraries, respectively.
+∗ sergi.julia@icfo.eu
+† alexandre.dauphin@icfo.eu
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+and
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+single atoms in a Hubbard-regime optical lattice,” Na-
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+ITensor Software Library for Tensor Network Calcula-
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+
+Supplemental Materials: Topological stripe state in an extended Fermi-Hubbard
+model
+Sergi Juli`a-Farr´e,1, ∗ Lorenzo Cardarelli,2, 3 Maciej Lewenstein,1, 4 Markus M¨uller,2, 3 and Alexandre Dauphin1, †
+1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,
+Av.
+Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
+2Peter Gr¨unberg Institute, Theoretical Nanoelectronics,
+Forschungszentrum J¨ulich, D-52428 J¨ulich, Germany
+3Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
+4ICREA, Pg.
+Llu´ıs Companys 23, 08010 Barcelona, Spain
+SI.
+DEFINITION OF THE QUANTUM ANOMALOUS HALL LOCAL ORDER PARAMETER
+In order to compute the local order parameter of the QAH phase ξQAH = �
+ij∈plaquette Im
+�
+ˆc†
+i ˆcj
+�
+, introduced in the
+main text, we need to follow the arrow convention shown in Fig. S1 for the bond i → j in each of the bulk plaquettes.
+FIG. S1. Depiction of the checkerboard lattice with the arrow convention for computing the current loop order parameter.
+SII.
+DETAILS ON THE DMRG CALCULATIONS
+The DMRG (iDMRG) calculations are performed with the iTensor [? ] and TeNPy [? ] libraries. In both cases, we
+use MPS bond dimensions up to χmax = 4000 to ensure truncation errors in the final MPS of εtrunc ∼ 10−5 at most.
+In order to find the QAH phase, which spontaneously breaks time-reversal symmetry, we add to the Hamiltonian
+a small complex field in each closed loop of NN, thus modifying the hopping strength t. That is, during the first
+sweeps of the DMRG algorithm, we add to the Hamiltonian a guiding field proportional to the QAH order parameter,
+i.e., of the form ih �
+ij∈plaq. ˆc†
+i ˆcj, for bonds i → j following the arrow convention of Fig. S1, and with h/t = 10−2.
+With this procedure, one can target solutions with spatially inhomogeneous patterns of the QAH order parameter
+by imprinting such patterns in a spatially dependent guiding field. However, as the DMRG algorithm is in general
+not able to escape local minima induced by the spatial pattern of such a guiding field, it is crucial to compare the
+final energies of different initial patterns. For instance, the topological stripe state discussed in the main text can be
+stabilized with the DMRG algorithm at half filling, but it is a metastable state. That is, its energy is larger than the
+spatially inhomogeneous QAH state. On the contrary, the topological stripe state is the ground state of the system
+in the hole-doped case.
+∗ sergi.julia@icfo.eu
+† alexandre.dauphin@icfo.eu
+arXiv:2301.03312v1  [cond-mat.quant-gas]  9 Jan 2023
+
+2
+SIII.
+MEAN-FIELD ANALYSIS OF THE LAUGHLIN PUMP
+Here we analyze the Laughlin pump of our interacting model within the mean-field treatment. Even though the
+DMRG method is more accurate, the single-particle nature of the mean-field ansatz allows for a better understanding
+of the quantized transfer of charges, in terms of the spectral flow of mid-gap localized states.
+For the mean-field analysis, we consider the same cylinder geometry and Hamiltonian of the main text, in which
+the free and interacting part read
+ˆH0 = − t
+�
+⟨ij⟩
+(ˆc†
+i ˆcj + H.c.) + J
+�
+⟨⟨ij⟩⟩
+eiφij(ˆc†
+i ˆcj + H.c.),
+(S1)
+and
+ˆHint = V1
+�
+⟨ij⟩
+ˆn′
+iˆn′
+j + V2
+�
+⟨⟨ij⟩⟩
+ˆn′
+iˆn′
+j + V3
+�
+⟨⟨⟨ij⟩⟩⟩
+ˆn′
+iˆn′
+j,
+(S2)
+with ˆn′
+i ≡ ˆni − 1/2 and ni = c†
+ici. We perform a standard Hartree-Fock decoupling of the density-density interaction
+terms
+ˆniˆnj ≃ −ξijˆc†
+jˆci − ξ∗
+ijˆc†
+i ˆcj + |ξij|2 + ¯niˆnj + ¯njˆni − ¯ni¯nj,
+(S3)
+with ξij ≡ ⟨ˆc†
+i ˆcj⟩ and ¯ni ≡ ⟨ˆni⟩. Notice that, within this approximation, the Hamiltonian becomes quadratic in the
+creation/annihilation fermionic operators. The ground state can then be found by iteratively diagonalizing the Hamil-
+tonian with a Bogoliubov transformation, in order to determine the mean-field values ξij and ¯ni in a self-consistent
+loop. From previous studies [? ? ], it is known that such mean-field ansatz can qualitatively capture the QAH phase
+of the system, with a shift in the interaction parameters in which the QAH phase appears, compared to the DMRG
+method. Therefore, here we change the interaction values used in the main text to V1 = 2.5t, V2 = 1.5t, and V3 = 0,
+which leads to a QAH phase at half filling in the mean-field ansatz.
+A.
+Homogeneous QAH phase
+For simplicity, let us start by analyzing the Laughlin pump in the homogeneous QAH phase found at half filling.
+We fix the mean-field parameters found self-consistently at the flux θ = 0, and investigate the response of the system
+under the flux insertion, as shown in Fig. S2. Figure S2(a) shows the mean-field single-particle energies of ˆH(θ) as
+a function of the inserted flux in the cylinder θ, with the color code indicating their center-of-mass position. Notice
+that the spectrum of ˆH(0) = ˆH(2π) is the same, and therefore the spectral flow of the left and right localized mid-gap
+states needs to be compensated by the bulk states. This leads to the charge flow from left to right, related to the
+presence of a nontrivial Chern number, which is shown in Figure S2(b). In contrast with the DMRG results, here we
+observe discrete jumps in the quantities ∆Q(θ), resulting in ∆Q(2π) − ∆Q(0) = 0 for both left and right edges. That
+is, even though the integrated slope of these quantities is quantized to ±1, no net charge transfer is observed. This is
+due to the breaking of adiabaticity in the single-particle method that we use, as shown in Figs. S2(c)-(d). Figure S2(c)
+shows the populated single-particle states during the pump. At the level crossing, and due to the fact that only half
+of the states are occupied at half filling, the right-localized edge state becomes unpopulated when crossing the Fermi
+energy, and the left-localized edge states get populated instead. This leads to a vanishing overlap between consecutive
+wavefunctions, |Ψ(θ)⟩ , |Ψ(θ + δθ)⟩, as can be seen in Fig. S2(d). Indeed, one can rigorously relate the value of these
+jumps to the topological invariant, as discussed in Ref. [? ]. It is worth noting that, in the DMRG results presented
+in the main text, such jumps are absent due to the local nature of the state optimization within this method, and the
+fact that at each value of the flux θ we initialize the DMRG algorithm with the previous converged state at θ − δθ.
+Interestingly, this local nature of the state update resembles the situation that one would encounter in a real adiabatic
+time evolution under a flux insertion. Alternatively, for random initial states, we also observe jumps with the DMRG
+method (not shown here).
+
+3
+(a)
+(b)
+(c)
+(d)
+FIG. S2. Mean-field results for a topological Laughlin pump in a cylinder with a spatially homogeneous interaction-induced
+QAH phase. The different quantities are plotted as a function of the inserted flux θ. (a) Single-particle mean-field spectrum.
+Here the color code indicates the center-of-mass position of each state. (b) The net charge on each half of the cylinder. (c)
+Same as in (a) but showing only the populated states. (d) Overlap between consecutive wavefunctions.
+B.
+Topological stripe state
+The mean-field Laughlin pump in the topological stripe state can be considered as a generalization of the results
+presented in Fig. S2. In accordance with the DMRG results presented in the main text, the mean-field method in
+the cylinder geometry also leads to a domain wall pattern of the QAH phase for a single hole added to half filling.
+The response of this system to a periodic flux insertion is shown in Fig. S3. Figure S3(a) shows the appearance of
+additional midgap states, compared to the case of a homogeneous QAH phase of the previous Fig. S2. Such additional
+states are localized at the center domain wall of the cylinder and are a direct signature of the difference in the Chern
+number between the left and right halves of the cylinder. This difference in Chern numbers also leads to an opposite
+direction of the charge flow on each half of the cylinder, as shown in Fig. S3(b), where one can observe that the charge
+flows from the left and right edges to the center, leading to a net accumulation in the center of the system. As in
+the homogeneous QAH phase, here we also observe jumps in the transferred charges, which are caused by the level
+crossings and vanishing overlaps between consecutive wavefunctions [see Figs. S3(c)-(d)].
+
+4
+(a)
+(b)
+(c)
+(d)
+FIG. S3. Mean-field results for a topological Laughlin pump in a cylinder with a spatially inhomogeneous interaction-induced
+QAH phase (topological stripe state). The different quantities are plotted as a function of the inserted flux θ. (a) Single-particle
+mean-field spectrum. Here the color code indicates the center-of-mass position of each state. (b) The net charge on the left,
+center, and right regions of the cylinder. (c) Same as in (a) but showing only the populated states. (d) Overlap between
+consecutive wavefunctions.
+

89FQT4oBgHgl3EQf5Db_/content/tmp_files/2301.13434v1.pdf.txt

@@ -0,0 +1,1696 @@
+MNRAS 000, 1–14 (2015)
+Preprint 1 February 2023
+Compiled using MNRAS LATEX style file v3.0
+DESI and DECaLS (D&D): galaxy-galaxy lensing measurements with 1%
+survey and its forecast
+Ji Yao1,2,3★ , Huanyuan Shan1,4† , Pengjie Zhang2,3,5‡, Eric Jullo6 , Jean-Paul Kneib6,7, Yu Yu2,3 ,
+Ying Zu2,3 , David Brooks8, Axel de la Macorra9, Peter Doel8, Andreu Font-Ribera10 ,
+Satya Gontcho A Gontcho11 , Theodore Kisner11 , Martin Landriau11 , Aaron Meisner12 ,
+Ramon Miquel13,10, Jundan Nie14 , Claire Poppett11,15,16, Francisco Prada17 , Michael Schubnell18,19,
+Mariana Vargas Magana9, and Zhimin Zhou14
+1Shanghai Astronomical Observatory (SHAO), Nandan Road 80, Shanghai, China
+2Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China
+3Key Laboratory for Particle Astrophysics and Cosmology (MOE)/Shanghai Key Laboratory for Particle Physics and Cosmology, China
+4 University of Chinese Academy of Sciences, Beijing, China
+5Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, China
+6Aix-Marseille Univ, CNRS, CNES, LAM, Marseille, France
+7Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
+8Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK
+9Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México
+10Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain
+11Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
+12NSF’s NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA
+13Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain
+14National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd., Chaoyang District, Beijing, 100012, P.R. China
+15Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way, Berkeley, CA 94720, USA
+16University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720, USA
+17Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain
+18Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
+19University of Michigan, Ann Arbor, MI 48109, USA
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+The shear measurement from DECaLS (Dark Energy Camera Legacy Survey) provides an excellent opportunity for galaxy-
+galaxy lensing study with DESI (Dark Energy Spectroscopic Instrument) galaxies, given the large (∼ 9000 deg2) sky overlap. We
+explore this potential by combining the DESI 1% survey and DECaLS DR8. With ∼ 106 deg2 sky overlap, we achieve significant
+detection of galaxy-galaxy lensing for BGS and LRG as lenses. Scaled to the full BGS sample, we expect the statistical errors
+to improve from 18(12)% to a promising level of 2(1.3)% at 𝜃 > 8
+′(< 8
+′). This brings stronger requirements for future
+systematics control. To fully realize such potential, we need to control the residual multiplicative shear bias |𝑚| < 0.01 and
+the bias in the mean redshift |Δ𝑧| < 0.015. We also expect significant detection of galaxy-galaxy lensing with DESI LRG/ELG
+full samples as lenses, and cosmic magnification of ELG through cross-correlation with low-redshift DECaLS shear. If such
+systematical error control can be achieved, we find the advantages of DECaLS, comparing with KiDS (Kilo Degree Survey)
+and HSC (Hyper-Suprime Cam), are at low redshift, large-scale, and in measuring the shear-ratio (to 𝜎𝑅 ∼ 0.04) and cosmic
+magnification.
+Key words: weak lensing – cosmology – galaxy-galaxy lensing
+1 INTRODUCTION
+Weak gravitational lensing is one of the most promising cosmological
+probes in studying the nature of dark matter, dark energy, and gravity
+★ E-mail: ji.yao@outlook.com (JY)
+† E-mail: hyshan@shao.ac.cn (HS)
+‡ E-mail: zhangpj@sjtu.edu.cn (PZ)
+(Refregier 2003; Mandelbaum 2018). The combination between dif-
+ferent probes can be even more powerful, due to more constraining
+power and breaking the degeneracy between the parameters (Planck
+Collaboration et al. 2020; DES Collaboration et al. 2021). However,
+possibly due to residual systematics or new physics beyond the stan-
+dard ΛCDM model, the tension between CMB (cosmic microwave
+background) at redshift 𝑧 ∼ 1100 and the late-time galaxy surveys at
+𝑧 <∼ 1 troubles us when using their synergy (Hildebrandt et al. 2017;
+© 2015 The Authors
+arXiv:2301.13434v1  [astro-ph.CO]  31 Jan 2023
+
+2
+J. Yao et al.
+Hamana et al. 2020; Hikage et al. 2019; Asgari et al. 2021; Heymans
+et al. 2021; DES Collaboration et al. 2021; Secco et al. 2022; Amon
+et al. 2021; Planck Collaboration et al. 2020). Many attempts have
+been made to examine this tension, in terms of different systematics
+(Yamamoto et al. 2022; Wright et al. 2020; Yao et al. 2020, 2017;
+Kannawadi et al. 2019; Pujol et al. 2020; Mead et al. 2021; Secco
+et al. 2022; Amon et al. 2022; Fong et al. 2019), different statistics
+(Asgari et al. 2021; Joachimi et al. 2021; Lin & Ishak 2017; Harnois-
+Déraps et al. 2021; Shan et al. 2018; Sánchez et al. 2021; Leauthaud
+et al. 2022; Chang et al. 2019), and possible new physics (Jedamzik
+et al. 2021). We also refer to recent reviews for the readers’ references
+(Perivolaropoulos & Skara 2021; Mandelbaum 2018).
+To fully understand the physics behind this so-called “𝑆8” tension,
+different cosmological probes are required, as their sensitivities to
+the systematics are different. Many new observations are also needed,
+to explore different redshift ranges, sky patches, and even equipment
+properties. Among the many proposed stage IV galaxy surveys like
+Dark Energy Spectroscopic Instrument (DESI DESI Collaboration
+et al. (2016a,b)), Vera C. Rubin Observatory’s Legacy Survey of
+Space and Time (LSST, LSST Science Collaboration et al. 2009),
+Euclid (Laureijs et al. 2011), Roman Space Telescope (or WFIRST,
+Spergel et al. 2015) and China Space Station Telescope (CSST, Gong
+et al. 2019), DESI is the only one currently operating and has mea-
+sured more than 7.5 million redshifts so far.
+DESI itself will provide tremendous constraining power in study-
+ing the expansion history of the Universe as well as the large-scale
+structure (DESI Collaboration et al. 2016a). Its cross-correlations
+with other lensing surveys (referred to as galaxy-galaxy lensing or
+g-g lensing) will provide not only more, but also independent cos-
+mological information (Prat et al. 2021; Joudaki et al. 2018; Sánchez
+et al. 2021), while it can be used to study the galaxy-matter relation
+(Leauthaud et al. 2022, 2017), test gravity (Zhang et al. 2007; Jullo
+et al. 2019; Blake et al. 2020), and study the systematics (Yao et al.
+2020, 2017; Zhang 2010; Zhang et al. 2010; Giblin et al. 2021).
+However, stage III surveys like DES (Dark Energy Survey, DES Col-
+laboration et al. 2021), KiDS (Kilo-Degree Survey, Heymans et al.
+2021), and HSC (Hyper-Suprime Cam, Hikage et al. 2019) do not
+offer extremely large overlap with DESI, while the stage IV surveys
+mentioned previously will require many years of observations before
+reaching their full overlap with DESI. In short, the sky overlap will
+limit the cross-correlation studies with DESI in the near future.
+In this work, we study the cross-correlations between galaxy shear
+measured from DECaLS (Dark Energy Camera Legacy Survey) DR8
+and galaxies from the DESI 1% (SV3) survey, and compare those
+with the overlapped data from KiDS and HSC. We measure the g-
+g lensing signals of the different weak lensing surveys with DESI
+1% survey and estimate their S/N (signal-to-noise ratio) that can be
+achieved with full DESI in the future. We explore the advantages of
+DECaLS, and exhibit the measurements of shear-ratio and cosmic
+magnification as two promising tools in using the great constraining
+power of DECaLS × DESI. Additionally, to achieve the expected
+precision, we propose requirements on the DECaLS data, in terms
+of the shear calibration and the redshift distribution calibration.
+This work is organized as follows. In Section 2 we briefly intro-
+duce the observables and their theoretical predictions. In Section 3
+we describe the DESI, DECaLS, KiDS, and HSC data we use. In
+Section 4 we show the g-g lensing measurements for different DESI
+density tracers and different lensing surveys, and the measurements
+of shear-ratio and cosmic magnification. We summarize our findings
+from DESI×DECaLS for the 1% survey in Section. 5.
+2 THEORY
+In this section, we briefly review the theory of the g-g lensing ob-
+servables. We assume spacial curvature Ω𝑘 = 0 so that the comoving
+radial distance equals the comoving angular diameter distance.
+2.1 Galaxy-galaxy lensing
+Since the foreground gravitational field can distort the shape of the
+background galaxy, there will be a correlation between the back-
+ground galaxies’ gravitational shear 𝛾G and the foreground galaxies’
+number density 𝛿g. The correlation of
+�
+𝛿g𝛾G�
+(or 𝑤gG) will probe the
+clustering of the underlying matter field ⟨𝛿m𝛿m⟩ (or the matter power
+spectrum 𝑃𝛿(𝑘)), the galaxy bias 𝑏𝑔(𝑘, 𝑧), and the redshift-distance
+relation, which are sensitive to the cosmological model and gravita-
+tional theory. We recall the g-g lensing angular power spectrum (Prat
+et al. 2021):
+𝐶𝑔𝜅 (ℓ) =
+∫ 𝜒max
+0
+𝑛l(𝜒)𝑞s(𝜒)
+𝜒2
+𝑏g(𝑘, 𝑧)𝑃𝛿
+�
+𝑘 = ℓ + 1/2
+𝜒
+, 𝑧
+�
+𝑑𝜒,
+(1)
+which is a weighted projection from the 3D non-linear matter power
+spectrum 𝑃𝛿(𝑘, 𝑧) to the 2D galaxy-lensing convergence angular
+power spectrum 𝐶𝑔𝜅 (ℓ). It will also depend on the galaxy bias 𝑏g =
+𝛿g/𝛿m, the comoving distance 𝜒, the redshift distribution of the
+lens galaxies 𝑛l(𝜒) = 𝑛l(𝑧)𝑑𝑧/𝑑𝜒, and the lensing efficiency as a
+function of the lens position (given the distribution of the source
+galaxies) 𝑞s(𝜒), which is written as
+𝑞s(𝜒l) = 3
+2Ωm
+𝐻2
+0
+𝑐2 (1 + 𝑧l)
+∫ ∞
+𝜒l
+𝑛s(𝜒s) (𝜒s − 𝜒l)𝜒l
+𝜒s
+𝑑𝜒s,
+(2)
+where 𝑛s(𝜒s) denotes the distribution of the source galaxies as a
+function of comoving distance, while 𝜒s and 𝜒l denote the comoving
+distance to the source and the lens, respectively.
+The real-space galaxy-shear correlation function can be obtained
+through the Hankel transformation
+𝑤gG(𝜃) = 1
+2𝜋
+∫ ∞
+0
+𝑑ℓℓ𝐶𝑔𝜅 (ℓ)𝐽2(ℓ𝜃),
+(3)
+where 𝐽2(𝑥) is the Bessel function of the first kind with order 2.
+The “G” represents the gravitational lensing shear 𝛾G, which is con-
+ventionally used to separate from the intrinsic alignment 𝛾I, whose
+contribution is ignored in this work due to the photo-𝑧 separation
+shown later.
+Therefore, by observing the correlation of 𝑤gG, we can derive the
+constraints on the cosmological parameters through Eq. (1), 𝑃𝛿(𝑘)
+and 𝜒(𝑧). In order to get the precise cosmology, many systemat-
+ics need to be considered, for example, the shear calibration error
+that can shift the measurement of 𝑤gG, the inaccurate estimation of
+redshift distribution for the source 𝑛s(𝜒s(𝑧s)) which can bias the
+theoretical estimation of Eq. (1), the massive neutrino effects and the
+baryonic effects that can bias the matter power spectrum 𝑃𝛿(𝑘, 𝑧),
+and the non-linear galaxy bias 𝑏𝑔(𝑘, 𝑧)1. In this work, we mainly
+focus on the statistical significance for DESI×DECaLS, rather than
+the systematics. The current statistical error for the 1% survey is
+expected to be more dominant, but for cautious reasons, we will not
+give final estimations on the cosmological parameters.
+1 In this work we use the mathematical classification of linear/non-linear
+bias as a matched filter, however, for more physical modeling, this is normally
+expressed as 1-halo/2-halo terms and HOD (halo occupation distribution)
+descriptions such as central/satellite fractions (Leauthaud et al. 2017)
+MNRAS 000, 1–14 (2015)
+
+D&D 1%
+3
+2.2 Shear-ratio
+The g-g lensing two-point statistics normally contain stronger de-
+tection significance at the small-scale than at the large-scale, due to
+a stronger tidal gravitational field and more galaxy pairs (through-
+out the whole sky, not around a particular galaxy). However, due
+to the inaccurate modeling of small-scale effects, such as the non-
+linear galaxy bias 𝑏g(𝑘, 𝑧), suppression in the matter power spectrum
+𝑃𝛿(𝑘) due to massive neutrino and baryonic effects, etc., the small-
+scale information is conventionally abandoned (Heymans et al. 2021;
+DES Collaboration et al. 2021; Lee et al. 2022). However, by choos-
+ing the same lens galaxies with source galaxies at different redshifts,
+i.e. with the same redshift distribution 𝑛𝑢(𝑧) for the lens while dif-
+ferent redshift distribution 𝑛𝑣 (𝑧) and 𝑛𝑤 (𝑧) for the sources, the ratio
+between the angular power spectra 𝐶𝑔𝜅
+𝑢𝑣 and 𝐶𝑔𝜅
+𝑢𝑤 (or the correla-
+tion functions 𝑤gG
+𝑢𝑣 and 𝑤gG
+𝑢𝑤) will mainly base on the two lensing
+efficiency functions as in Eq. (2) for the 𝑣-th and 𝑤-th source bins.
+This ratio does not suffer strongly from the modeling of the galaxy
+bias 𝑏g or the matter power spectrum 𝑃𝛿(𝑘), as they share the same
+lens sample according to Eq. (1). The shear-ratio (or lensing-ratio)
+has been used to improve cosmological constraints (Sánchez et al.
+2021), as it is sensitive to the 𝜒(𝑧) relation in Eq. (2) and the nuisance
+parameters for the systematics, or to study the shear bias (Giblin et al.
+2021). In this work, we will show the great potential of measuring
+shear-ratio with DESI×DECaLS.
+To account for the full covariance in measuring shear-ratio 𝑅 =
+𝑤2/𝑤1, and to prevent possible singular values when taking the ratio
+(when 𝑤1 ∼ 0), we construct the following data vector
+𝑉 = 𝑤1𝑅 − 𝑤2,
+(4)
+which is designed to be 0 when 𝑅 is correctly predicted from the
+two data sets 𝑤1 and 𝑤2 that we want to take the ratio. The resulting
+covariance for the data vector 𝑉 is
+𝐶′ = 𝑅2𝐶11 + 𝐶22 − 𝑅(𝐶12 + 𝐶21),
+(5)
+where 𝐶𝑖 𝑗 is the covariance between 𝑤𝑖 and 𝑤 𝑗. The likelihood
+of −2lnℒ = 𝑉T𝐶′−1𝑉 will give the posterior of the shear-ratio 𝑅.
+To account for the covariance is 𝑅-dependent, normalization is done
+thereafter so that its PDF satisfies
+∫
+𝑃(𝑅)𝑑𝑅 = 1. An alternative way
+is to marginalize over the theoretical predictions 𝑤𝑖, similar to Sun
+et al. (2022); Dong et al. (2022), which we leave for future studies.
+2.3 Cosmic magnification
+The observed galaxy number density is affected by its foreground
+lensing signals, leading to an extra fluctuation besides the intrinsic
+clustering of galaxies, namely,
+𝛿L
+g = 𝛿g + 𝑔𝜇𝜅,
+(6)
+where 𝛿Lg denotes the observed lensed galaxy overdensity, 𝛿g denotes
+the intrinsic overdensity of galaxies due to gravitational clustering,
+𝜅 is the lensing convergence affecting the flux and the positions of
+the foreground galaxy sample, and due to the foreground inhomo-
+geneities. For a complete and flux-limited sample, the magnification
+amplitude 𝑔𝜇 = 2(𝛼 − 1). In that case, the magnification amplitude
+is sensitive to the galaxy flux function 𝑁(𝐹), denoting the number
+of galaxies brighter than flux limit 𝐹, with 𝛼 = −𝑑ln𝑁/𝑑ln𝐹.
+According to Eq. (6), for a given galaxy sample at 𝑧 = 𝑧1, it not
+only contains clustering information of 𝛿g(𝑧 = 𝑧1), but also has
+lensing information of 𝜅 from the matter at 𝑧 < 𝑧1, which is normally
+treated as a contamination to the clustering signals (von Wietersheim-
+Kramsta et al. 2021; Deshpande & Kitching 2020; Kitanidis & White
+2021). Meanwhile, attempts have been made to directly measure the
+cosmic magnification as a source of cosmological information (Liu
+et al. 2021; Gonzalez-Nuevo et al. 2020; Yang et al. 2017).
+We follow the method of Liu et al. (2021) and correlate the shear
+galaxies at lower redshift (bin 𝑖) and the number density galaxies at
+higher redshift (bin 𝑗),
+𝐶𝜅𝜇
+𝑖 𝑗 (ℓ) = 𝑔𝜇
+∫ 𝜒max
+0
+𝑞𝑖(𝜒)𝑞 𝑗 (𝜒)
+𝜒2
+𝑃𝛿
+�
+𝑘 = ℓ + 1/2
+𝜒
+, 𝑧
+�
+𝑑𝜒,
+(7)
+which requires the redshift distribution of 𝑛𝑖(𝑧) being significantly
+separated from 𝑛 𝑗 (𝑧), so that the intrinsic clustering × lensing shear
+signal vanishes. The corresponding correlation function from the
+Hankel transformation is similar to Eq. (3).
+2.4 Signal-to-noise definition
+The S/N definition in this work uses amplitude fitting. For a given
+measurement 𝑤data and an assumed theoretical model 𝑤model, we fit
+an amplitude 𝐴 to the likelihood:
+−2lnℒ = (𝑤data − 𝐴𝑤model) Cov−1 (𝑤data − 𝐴𝑤model) ,
+(8)
+so that a posterior of 𝐴+𝜎𝐴
+−𝜎𝐴 can be obtained, where 𝜎𝐴 is the Gaussian
+standard deviation. Then the corresponding S/N is 𝐴/𝜎𝐴.
+We note that, if 𝑤data is a single value rather than a data vector, this
+S/N defined by amplitude fitting is identical to the S/N of the data
+itself, namely 𝐴/𝜎𝐴 = 𝑤data/𝜎𝑤data. This is the case for most of the
+S/N calculated in this work, when there is one single measurement
+at small-scale and one at large-scale, and the small-scale and large-
+scale data correspond to different (nonlinear/linear) galaxy biases so
+they should be treated separately.
+3 DATA
+In this section, we introduce the DESI spectroscopic data and the
+shear catalogs from DECaLS/KiDS/HSC. We note even though the
+DES-Y3 catalog can have an overlap with full DESI for ∼ 1264 deg2,
+its overlap with DESI SV3 catalog is 0. We, therefore, do not present
+any analysis for DES.
+3.1 DESI
+DESI is the only operating Stage IV galaxy survey. It is designed
+to cover 14,000 deg2 of the sky, with 5,000 fibers collecting spectra
+simultaneously (DESI Collaboration et al. 2016b; Silber et al. 2022;
+Miller et al. 2022). DESI aims to observe density tracers such as BGS
+(Bright Galaxy Survey, Ruiz-Macias et al. 2020), LRG (luminous red
+galaxies, Zhou et al. 2020), ELG (emission line galaxies, Raichoor
+et al. 2020), and QSO (quasi-stellar objects, Yèche et al. 2020), with
+generally increasing redshift. Other supporting papers on target se-
+lections and validations can be find in Allende Prieto et al. (2020);
+Alexander et al. (2022); Lan et al. (2022); Cooper et al. (2022); Hahn
+et al. (2022); Zhou et al. (2022); Chaussidon et al. (2022). DESI
+plans to use these tracers to study cosmology, especially in BAO
+(baryonic acoustic oscillations) and RSD (redshift-space distortions)
+(DESI Collaboration et al. 2016a; Levi et al. 2013). It is located on
+the 4-meter Mayall telescope in Kitt Peak, Arizona (DESI Collabo-
+ration et al. 2022). From 2021 till now, DESI has finished its “SV3”
+(DESI collaboration et al. 2022) and “DA0.2” catalogs, which will
+be included in the coming Early Data Release (EDR, DESI collabo-
+ration et al. 2023). The Siena Galaxy Atlas (Moustakas et al. 2022)
+is also expected soon.
+MNRAS 000, 1–14 (2015)
+
+4
+J. Yao et al.
+The DESI experiment is based on the DESI Legacy Imaing Surveys
+(Zou et al. 2017; Dey et al. 2019; Schlegel et al. 2022), with multiple
+supporting pipelines in spectroscopic reduction (Guy et al. 2022),
+derivation of classifications and redshifts (Bailey et al. 2022), fiber
+assigement (Raichoor et al. 2022), survey optimization (Schlafly et
+al. 2022), spectroscopic target selection (Myers et al. 2022)
+In this work, we use the DESI SV3 catalog, which is also known
+as the 1% survey (with a sky coverage of ∼ 140 deg2), for the
+g-g lensing study. We consider the DESI BGS, LRGs, and ELGs,
+while ignoring the QSOs as the available number is relatively low.
+In SV3, each galaxy is assigned a weight to account for the survey
+completeness and redshift failure. Since the purpose of this paper
+is not a precise measurement of cosmology, we assume the linear
+galaxy biases follow 𝑏BGS(𝑧)𝐷(𝑧) = 1.34, 𝑏LRG(𝑧)𝐷(𝑧) = 1.7,
+and 𝑏ELG(𝑧)𝐷(𝑧) = 0.84, where 𝐷(𝑧) is the linear growth factor
+normalized to 𝐷(𝑧 = 0) = 1 (DESI Collaboration et al. 2016a). The
+number of galaxies used will be informed later in the paper, as the
+overlap between the DESI 1% survey and the lensing surveys are
+different.
+3.2 DECaLS
+We use lensing shear measurement from DECaLS DR8, which con-
+tains galaxy images in 𝑔−, 𝑟−, and 𝑧−bands (Dey et al. 2019). DE-
+CaLS DR8 galaxies are processed by Tractor (Meisner et al. 2017;
+Lang et al. 2014) and divided into five types according to their mor-
+phologies: PSF, SIMP, DEV, EXP, and COMP (Phriksee et al. 2020;
+Yao et al. 2020; Zu et al. 2021; Xu et al. 2021). The galaxy ellipticities
+𝑒1,2 are measured —- except for the PSF type —- with a joint fit on
+the 𝑔−, 𝑟−, and 𝑧−bands. A conventional shear calibration (Heymans
+et al. 2012; Miller et al. 2013; Hildebrandt et al. 2017) is applied as
+in
+𝛾obs = (1 + 𝑚)𝛾true + 𝑐,
+(9)
+with a multiplicative bias 𝑚 and additive bias 𝑐, to account for
+possible residual bias from PSF modeling, measurement method,
+blending and crowding (Mandelbaum et al. 2015; Euclid Collabo-
+ration et al. 2019). This calibration is obtained by comparing with
+Canada–France–Hawaii Telescope(CFHT) Stripe 82observed galax-
+ies and Obiwan simulated galaxies (Phriksee et al. 2020; Kong et al.
+2020).
+Several versions of the photometric redshift for the DECaLS galax-
+ies have been estimated (Zou et al. 2019; Zhou et al. 2021; Duncan
+2022). We apply the most widely used one (Zhou et al. 2021), which
+uses the 𝑔, 𝑟, and 𝑧 optical bands from DECaLS while borrowing
+𝑊1 and 𝑊2 infrared bands from WISE (Wide-field Infrared Survey
+Explorer, Wright et al. 2010). The photo-𝑧 algorithm is trained based
+on a decision tree, with training samples constructed from a wide
+selection of spectroscopic redshift surveys and deep photo-𝑧 surveys.
+We additionally require 𝑧 < 21 to select galaxies with better photo-𝑧.
+We use the photo-z distribution to represent the true-z distribution
+𝑛(𝑧), while allowing a systematic bias of Δ𝑧 in the form 𝑛(𝑧 − Δ𝑧),
+to pass its effect to Eq. (2) then Eq. (1). This is appropriate as weak
+lensing is mainly biased due to the mean redshift but slightly affected
+by the redshift scatter.
+Overall, the DR8 shear catalog has ∼ 9, 000 deg2 sky coverage —-
+which will be the final overlap with full DESI —- with an average
+galaxy number density of ∼ 1.9 gal/arcmin2. The overlapped area
+with DESI 1% survey is ∼ 106 deg2, which is significantly larger
+than the other stage III lensing surveys.
+We note that the current DECaLS DR8 shear catalog can have
+some residual multiplicative bias |𝑚| ∼ 0.05 (Yao et al. 2020; Phrik-
+see et al. 2020), possibly due to the selections in observational data
+while making the comparison (Li et al. 2020; Jarvis et al. 2016). This
+will prevent us from getting reliable cosmology for measurements
+with 𝑆/𝑁 >∼ 20. Also, there exists a possible bias in the redshift
+distribution 𝑛(𝑧), which will require a galaxy color-based algorithm
+(Hildebrandt et al. 2017; Buchs et al. 2019; Wright et al. 2020) or
+a galaxy clustering-based algorithm (Peng et al. 2022; Zhang et al.
+2010; van den Busch et al. 2020) to get the correction. For these
+two reasons, we choose not to extend this study to the precision cos-
+mology level. A future version of the DECaLS DR9 shear catalog
+is under development, with improved data reduction and survey pro-
+cedures2, with more advanced shear calibration for a pure Obiwan
+image simulation-based algorithm (Yao et al. in preparation) and
+redshift calibration (Xu et al. in preparation).
+3.3 KiDS
+The Kilo-Degree Survey is run by the European Southern Observa-
+tory and is designed for weak lensing studies in 𝑢𝑔𝑟𝑖 optical bands.
+The KiDS data are processed by THELI (Erben et al. 2013) and
+Astro-WISE (de Jong et al. 2015; Begeman et al. 2013). The galaxy
+shear measurements are obtained by 𝑙𝑒𝑛𝑠fit (Fenech Conti et al. 2017;
+Miller et al. 2013), and the photo-𝑧s are measured by BPZ (Benitez
+2000; Benítez et al. 2004) using the KiDS 𝑢𝑔𝑟𝑖 optical bands and
+the 𝑍𝑌𝐽𝐻𝐾s infrared bands from VIKING (Wright et al. 2019). The
+KiDS shears are calibrated following the same equation as Eq. (9)
+with image simulation Kannawadi et al. (2019).
+We use the KiDS-1000 shear catalog (Giblin et al. 2021; Asgari
+et al. 2021) in this work. The overlapped area with DESI SV3 is ∼ 55
+deg2. The expected overlapped area between the full DESI footprint
+and KiDS-1000 is ∼ 456 deg2.
+3.4 HSC
+The Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP, or
+HSC) is a Japanese lensing survey using the powerful Subaru tele-
+scope. It covers five photometric bands 𝑔𝑟𝑖𝑧𝑦. Compared with KiDS
+and DES, HSC has its unique advantage in the galaxy number density
+and high-z galaxies (but with a smaller footprint). The HSC shears
+are calibrated similarly to Eq. (9) (Mandelbaum et al. 2018) but with
+an additional shear responsivity (Hamana et al. 2020).
+We use the HSC-Y1 shear catalog (Hikage et al. 2019; Hamana
+et al. 2020), which overlaps with DESI SV3 for ∼ 48 deg2. The
+expected overlap between HSC-Y3 data and full DESI is ∼ 733 deg2.
+4 RESULTS
+In this section, we show the measurements of different galaxy-shear
+correlation functions. The estimator for the galaxy-shear correlation
+is:
+𝑤gG(𝜃) =
+�
+ED wE𝛾+
+EwD
+�
+ER(1 + 𝑚E)wEwR
+−
+�
+ER wE𝛾+
+EwR
+�
+ER(1 + 𝑚E)wEwR
+,
+(10)
+where wE, 𝑚E and 𝛾+
+E denotes the lensing weight (inverse-variance
+weight for DECaLS Phriksee et al. 2020 and HSC Hikage et al. 2019,
+an adjusted version for KiDS Miller et al. 2013), the multiplicative
+2 https://www.legacysurvey.org/dr9/description/
+MNRAS 000, 1–14 (2015)
+
+D&D 1%
+5
+Figure 1. The galaxy redshift distributions for the DESI BGS with 0 < 𝑧 <
+0.5 and photo-𝑧 distributions for the lensing surveys with 0.6 < 𝑧𝑝 < 1.5.
+The numbers in the labels are the number of galaxies in the overlapped region.
+bias correction (for HSC there is an extra shear responsivity in-
+cluded), and the tangential shear of the source galaxy, with respect to
+the given lens galaxy with weight wD or wR. The Σ-summations are
+calculated for all the ellipticity-density (ED) pairs and the ellipticity-
+random (ER) pairs. We note Eq. (10) already includes the correction
+for boost factor (Mandelbaum et al. 2005; Amon et al. 2018), and
+this equation is adequate for the multiplicative bias 𝑚E defined either
+per galaxy or per sample. The correlation uses DESI official ran-
+dom catalogs to simultaneously correct for the additive bias in the
+presence of a mask and reduce the shape noise. We will show the
+measurements with different lens samples and source catalogs using
+the above estimator.
+4.1 DESI 𝑤gG
+We first show the g-g lensing measurements for DESI BGS and the
+three shear catalogs. The normalized redshift distributions 𝑛(𝑧) are
+shown in Fig. 1, with the number of galaxies being used in the labels.
+We use BGS with 0 < 𝑧 < 0.5, and require the photo-𝑧 of the source
+galaxies located at 0.6 < 𝑧𝑝 < 1.5, so that the overlap in redshift
+is very small even considering the inaccuracy of photo-𝑧. We see
+that DECaLS has the most available BGS lenses, while HSC has
+the most available sources and the highest redshift. We notice there
+are unexpected spikes for the photo-z distribution of KiDS, which
+is probably due to cosmic variance as the overlapped area is much
+smaller than the full KiDS data.
+We show the measured correlation functions for the DESI BGS
+g-g lensing in Fig. 2. The correlations are measured in 2 logarithmic
+bins in 0.5 < 𝜃 < 80 arcmin, with the statistical uncertainties cal-
+Figure 2. The galaxy-galaxy lensing angular correlation functions, corre-
+sponding to the galaxies samples in Fig. 1. In the upper panel, the theo-
+retical curves are given by the fiducial cosmology and the assumed galaxy
+bias model. The {small-scale, large-scale} detection significances are {9.1,
+5.8} for BGS×DECaLS, {10.2, 3.9} for BGS×KiDS , and {16.1, 4.3} for
+BGS×HSC. In the lower panel, we show the ratio between our measurements
+and the corresponding theoretical model, with the latter re-weighted using the
+number of pairs and lensing weights to account for the band power problem
+with wide angular bins. The DECaLS and HSC results are slightly shifted
+horizontally.
+culated using jackknife re-sampling. We find that all three lensing
+surveys have strong g-g lensing signals, even for the current 1% DESI
+data. The measurements are shown in blue dots (DECaLS), orange
+triangles (KiDS), and green squares (HSC), while the corresponding
+theoretical comparisons are shown in the blue solid curve, the orange
+dash-dotted curve, and the green dotted curve. From this figure, we
+find that the advantage of DECaLS is its large-scale cosmological
+information, with the highest S/N ∼ 5.8. This is due to DECaLS’s
+significantly large overlap with DESI, reducing the cosmic variance.
+On the other hand, KiDS and HSC has larger S/N than DECaLS at
+small-scale, due to their higher source galaxy number density, which
+lowers the shape noise.
+In this work we choose not to estimate the best-fit cosmology, as for
+DECaLS, there are some unaddressed potential systematics (as dis-
+cussed in Sec 3.2), while for KiDS and HSC we do not want to harm
+the ongoing blinding efforts in the DESI collaboration (although for
+a larger catalog with the larger overlapped area). The theoretical es-
+timations in Fig. 2 and all the other similar figures in this work are
+based on the KiDS-1000 COSEBI ΛCDM cosmology with max-
+imum posterior of the full multivariate distribution (MAP, Asgari
+et al. (2021)), which has ℎ = 0.727, Ωbℎ2 = 0.023, Ωcℎ2 = 0.105,
+𝑛s = 0.949 and 𝜎8 = 0.772. We note the choice of other fiducial
+cosmology (Planck Collaboration et al. 2020; Asgari et al. 2021;
+DES Collaboration et al. 2021; Hamana et al. 2020) will give similar
+results for the current stage with DESI SV3. The linear galaxy biases
+are assumed following the descriptions of difference density tracers
+in Sec 3.1.
+We note that the choice of 2 log-bins is limited by the 20 jack-
+knife sub-regions (Yao et al. 2020; Mandelbaum et al. 2006), which
+MNRAS 000, 1–14 (2015)
+
+4
+BGS 132688
+DFCaLS 132484
+3
+N
+n
+2
+1
+0
+BGS71187
+3
+KiDS.781045
+N
+n
+1
+BGS 63381
+3
+HSC 2008890
+N
+1
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Z or Zp10-3
+wgG(0)
+BGS DECaLS
+BGS KiDS
+BGS HSC
+BGS DECaLS
+BGS KiDS
+BGS HSC
+/ model
+1.5
+data /
+1.0
+100
+101
+θ [arcmin]6
+J. Yao et al.
+Figure 3. The galaxy-galaxy lensing angular correlation function 𝑤gG (upper
+panel) and its 45 deg-rotation test 𝑤gX (lower panel) for the BGS×DECaLS
+g-g lensing only, with the same distribution as in Fig. 1 but with more angular
+bins with 50 jackknife sub-regions. In the upper panel, the theoretical curves
+are given by the fiducial cosmology and the assumed galaxy bias model.
+The detection significance for the 5 angular bins are {6.5, 6.6, 8.4, 4.7, 3.2},
+with the 4 large-scale bins well-agreed with the prediction from fiducial
+cosmology and the linear bias assumption. The total S/N using amplitude
+fitting (as described in Sec. 2.4) is 8.9𝜎 (𝐴 = 1.03+0.12
+−0.11) for the right three
+large-scale dots, and is 10.0𝜎 (𝐴 = 1.0+0.1
+−0.1) for the right four large-scale
+dots. In the lower panel where the shear are rotated for 45 deg, the results are
+consistent with 0, with reduced-𝜒2 ∼ 3/5.
+is limited by: (1) the requirement of each jackknife sub-region is
+independent up to the largest scale we use (80 arcmin), and (2) the
+size of the overlapped region for KiDS and HSC (∼ 50 deg2). As the
+DESI survey expands, the available overlapped region will increase
+accordingly, resulting in increases in both the available number of
+sub-regions and the maximum angular scale we can measure. Alter-
+natively, we can use an analytical covariance (similar to Appendix
+A but more tests need to be done) or simulation based covariance
+for future DESI data. We also note in this work the inverses of the
+covariances are corrected (Hartlap et al. 2007; Wang et al. 2020) due
+to the limited number of sub-regions.
+As
+a
+demonstration
+of
+more
+angular
+binning,
+we
+use
+BGS×DECaLS data to show the choice of 50 jackknife sub-regions
+and 5 angular bins, as in Fig. 3. We show that with proper binning,
+more cosmological information can be extracted. The 𝜃 >∼ 2 arcmin
+measurements (the right 4 large-scale dots) agree with the linear bias
+assumption very well. In the future, with a larger overlapped foot-
+print, more jackknife sub-regions can be used, so that more angular
+bins can be measured, either to increase the total S/N or to address
+any scale-dependent systematics. We do see great potential for DE-
+CaLS from the above results, although measurements will ultimately
+be limited by systematic errors.
+We show the redshift distribution of the DESI LRGs and the three
+lensing surveys in Fig. 4, requiring 𝑧 < 0.6 for the spec-𝑧 LRGs
+and 0.7 < 𝑧𝑝 < 1.5 for the source galaxies. Similar to the BGS,
+more LRGs can be used when overlapping with DECaLS, while the
+available DECaLS source galaxies are less than in the other surveys.
+Figure 4. The galaxy redshift distributions for the DESI LRGs with 0 < 𝑧 <
+0.6 and photo-𝑧 distributions for the lensing surveys with 0.7 < 𝑧𝑝 < 1.5.
+The numbers in the labels are the number of galaxies in the overlapped region.
+Figure 5. The galaxy-galaxy lensing angular correlation functions, corre-
+sponding to the galaxies samples in Fig. 4. In the upper panel, the theo-
+retical curves are given by the fiducial cosmology and the assumed galaxy
+bias model. The {small-scale, large-scale} detection significances are {3.5,
+1.9} for LRG×DECaLS, {8.7, 2.2} for LRG×KiDS, and {10.6, 2.4} for
+LRG×HSC.
+MNRAS 000, 1–14 (2015)
+
+BGS DECaLS
+10-3
+10-4
+1
+5
+100
+101
+θ[arcmin]5
+LRG 18825
+4
+DECaLS78222
+3
+2
+1
+0
+5
+LRG 10542
+4
+KiDS 566019
+N
+3
+n
+2
+1
+05
+LRG 9230
+4
+HSC 1674031
+N
+3
+n
+2
+1
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.0
+1.4
+1.6
+z or Zp10-3
+LRG DECaLS
+LRG KiDS
+LRG HSO
+LRG DECaLS
+LRG KiDS
+10-4
+LRG HSO
+data / model
+2
+100
+101
+θ[arcmin]D&D 1%
+7
+Figure 6. The galaxy redshift distributions for the DESI ELGs with 0 < 𝑧 <
+0.7 and photo-𝑧 distributions for the lensing surveys with 0.8 < 𝑧𝑝 < 1.5.
+The numbers in the labels are the number of galaxies in the overlapped region.
+Since LRGs are generally distributed at higher 𝑧 than the BGS, we
+choose to increase the 𝑧-cut of the LRGs and the 𝑧𝑝-cut of the
+sources, resulting in reduced source galaxies compared with Fig. 1.
+This figure shows the DECaLS source galaxies are more reduced
+(from 133k to 78k) as it is shallower than the other two.
+The correlation measurements for the LRGs are presented in Fig. 5.
+At large-scale, the DECaLS signal is weaker than KiDS and HSC, but
+it still offers comparable S/N. At the small-scale, the S/N is dominated
+by deep surveys. The small-scale measurements are significantly
+higher than the theoretical predictions, due to LRGs being generally
+more massive than BGS, with stronger non-linear galaxy bias at such
+separations.
+Furthermore, we study the g-g lensing measurements of the DESI
+ELGs. We show the redshift distribution of the DESI ELGs and the
+three lensing surveys in Fig. 6, requiring 𝑧 < 0.7 for the spec-𝑧 ELGs
+and 0.8 < 𝑧𝑝 < 1.5 for the source galaxies. The available number of
+galaxies is further reduced compared to BGS and LRGs, due to DESI
+ELGs being mainly distributed at 𝑧 > 0.7. And the high-z sources
+for DECaLS are significantly less than KiDS and HSC.
+The correlation measurements of the ELGs are shown in Fig. 7.
+HSC appears to have the largest S/N at both large-scale and small-
+scale, and the S/N of DECaLS at large-scale is comparable to KiDS.
+All three lensing surveys have small-scale measurements lower than
+the theoretical predictions, suggesting the low measurement is not
+a systematics of DECaLS. We suspect this might be due to shape
+noise, sample variance, or possibly non-linear galaxy bias. As when
+we go from large-scale to small-scale, the non-linear halo bias for less
+massive halos (for example the host halos for ELGs, see Fig. 7) tends
+to drop compared with its linear bias, while the non-linear halo bias
+Figure 7. The galaxy-galaxy lensing angular correlation functions, corre-
+sponding to the galaxies samples in Fig. 6. The theoretical curves are given
+by the fiducial cosmology and the assumed galaxy bias model. The {small-
+scale, large-scale} detection significance are {-0.3, 1.4} for ELG×DECaLS,
+{-1.1, 1.4} for ELG×KiDS, and {2.5, 2.6} for ELG×HSC. The negative val-
+ues at small-scale represent negative measurements, which might be due to
+the non-linear galaxy bias, satellite fraction, or shot noise.
+tends to increase for the more massive halos (for example the host
+halos for the LRGs, see Fig. 5) according to Fig. 1 of Fong & Han
+(2021). The satellite galaxy fraction in the ELGs could also lead to a
+low amplitude at small-scale (Niemiec et al. 2017; Favole et al. 2016;
+Gao et al. 2022). These will require a higher S/N to test in the future.
+In this work, we only focus on large-scale ELGs measurement.
+4.2 Forecasts and Systematics
+We summarize our findings for the g-g lensing measurements from
+BGS (Fig. 2), LRGs (Fig. 5), and ELGs (Fig. 7) in Table 1. We see
+that DECaLS has its unique advantage in extracting cosmological in-
+formation at large-scale and at lower redshift (when correlating with
+the DESI BGS). Neglecting systematic errors for the moment, which
+will be dominant in practice, we give the forecast of the S/N with the
+complete DESI survey by re-scaling the covariance according to the
+overlapped area. This re-scaling assumes the covariance of the g-g
+lensing signal is dominated by the Gaussian covariance. Since we are
+extrapolating from small regions with significant boundary effects in
+our large-scale bin, this is only an approximation. We theoretically
+test the different components of the covariance in Appendix A for
+your interest. The large-scale information of future DECaLS×BGS
+can reach > 50𝜎, which is stronger than most of the current g-g
+lensing data, and will be very promising in studying the current 𝑆8
+tension (Hildebrandt et al. 2017; Hamana et al. 2020; Hikage et al.
+2019; Asgari et al. 2021; Heymans et al. 2021; DES Collaboration
+et al. 2021; Secco et al. 2022; Amon et al. 2021; Planck Collaboration
+et al. 2020). The contribution from LRGs and ELGs, and possibly
+QSOs in the future, can also offer independent cosmological infor-
+mation.
+We note that the S/N predictions in Table 1 ignored the potential
+MNRAS 000, 1–14 (2015)
+
+6
+ELG 10072
+DECaLS 44498
+4
+N
+n
+2
+05
+ELG 5296
+4 
+KiDS 433356
+N
+3
+n
+2
+1
+0
+4
+ELG 5213
+HSC 1409305
+3 
+N
+n
+2
+1
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Z or Zp10-3
+ELG DECaLS
+ELG KiDS
+10-4
+ELG HSC
+ELG DECaLS
+ELG KiDS
+ELG HSC
+data/ model
+2
+0
+100
+101
+θ[arcmin]8
+J. Yao et al.
+Table 1. We summarize the S/N of the DESI 1% survey (SV3) g-g lensing results in Fig. 2, 5 and 7, and forecast the ideal final S/N with full DESI, by rescaling the
+covariance based on the overlapped area, and assuming DECaLS data can be well calibrated. We note that the ELG measurements become negative sometimes,
+and therefore decide not to predict its final S/N. From this figure, we see that the advantage of DECaLS is at low-z (with BGS) and large-scale. We additionally
+present the possible bias in the forecasted S/N, namely ΔS/N. It includes the contribution from the statistical error of the current measurement, and residual
+systematical bias from the data calibration. We use multiplicative bias |𝑚| ∼ 0.05 (Yao et al. 2020; Phriksee et al. 2020) and redshift bias |Δ𝑧 | ∼ 0.02 (Zhou
+et al. 2021) for DECaLS DR8, |𝑚| ≤ 0.015 and |Δ𝑧 | ≤ 0.013 for KiDS (Asgari et al. 2021), and |𝑚| ≤ 0.03 and |Δ𝑧 | ≤ 0.038 for HSC (Hikage et al. 2019),
+to predict their systematical error in the forecasted S/N. We note the statistical contribution of ΔS/N results from rescaling the 1𝜎 error from Fig. 2, 5 and 7, and
+is scale-independent and redshift-independent. The contribution from multiplicative bias 𝑚 is also scale-independent, while the contribution from redshift bias
+Δ𝑧 is weakly scale-dependent and redshift-dependent. In the table, we only show the ΔS/N(Δ𝑧) values corresponding to the BGS results at the large-scale.
+survey
+SV3 overlap
+SV3 S/N [small-scale, large-scale]
+full overlap
+ideal forecast S/N [small-scale, large-scale]
+forecast potential bias ΔS/N
+[deg2]
+BGS
+LRG
+ELG
+[deg2]
+BGS
+LRG
+ELG
+statistical
+systematical
+DECaLS
+106
+[9.1, 5.8]
+[3.5, 1.9]
+[-0.3, 1.4]
+∼ 9000
+[83.8, 53.4]
+[32.2, 17.5]
+[N/A, 12.9]
+±9.2
+±5%(𝑚) ± 1.4%(Δ𝑧)
+KiDS
+55
+[10.2, 3.9]
+[8.7, 2.2]
+[-1.1, 1.4]
+456 (DR4)
+[29.3, 11.2]
+[25.1, 6.3]
+[N/A, 4.0]
+±2.9
+±1.5%(𝑚) ± 0.8%(Δ𝑧)
+HSC
+48
+[16.1, 4.3]
+[10.6, 2.4]
+[2.5, 2.6]
+733 (Y3)
+[62.9, 16.8]
+[41.4, 9.4]
+[9.8, 10.2]
+±3.9
+±3%(𝑚) ± 1.6%(Δ𝑧)
+Figure 8. The impact of the residual shear multiplicative bias 𝑚 and the bias in
+the redshift distribution Δ𝑧. For different 𝑚 and Δ𝑧, we evaluate the resulting
+𝑤bias/𝑤true at the large-scale of Fig. 2, 5 and 7 (𝜃 ∼ 51 arcmin) and show
+the ratio as the color map. The effect of 𝑚 is totally scale-independent, while
+the effect of Δ𝑧 is weakly scale-dependent, which can bring an additional
+∼ 20% difference at maximum. We also show where the bias from 𝑚 and Δ𝑧
+perfectly cancel each other (black solid curve), and the location where the net
+bias reaches ±0.01 (blue dashed curve) and ±0.02 (orange dotted curve).
+bias from systematics, such as residual shear multiplicative bias 𝑚
+and redshift distribution 𝑛(𝑧). The existence of the shear multiplica-
+tive bias 𝑚 will change the lensing efficiency from 𝑞s to (1 + 𝑚)𝑞s
+in Eq. (1) and (2). The bias in redshift distribution Δ𝑧 will change
+the redshift distribution for the source galaxies from 𝑛s(𝜒s(𝑧s)) to
+𝑛s(𝜒s(𝑧s − Δ𝑧)) in Eq. (2), so that the whole redshift distribution is
+shifted towards higher-z direction by Δ𝑧. For example, if we assume
+the residual multiplicative bias is |𝑚| ∼ 0.05 (which is found for
+some DECaLS galaxy sub-samples as in Phriksee et al. (2020); Yao
+et al. (2020)), and enlarge the covariance to account for this potential
+bias, then the S/N of DECaLS×BGS at large-scale will be reduced
+from > 50𝜎 to ∼ 20𝜎. This is a huge loss of cosmological informa-
+tion, although ∼ 20𝜎 is still comparable to the ∼ 11𝜎 of KiDS-DR4
+and ∼ 17𝜎 of HSC-Y3. Therefore, we emphasize the importance of
+calibrating DECaLS data in a more precise way in the future for reli-
+able cosmological measurements. We note the current measurements
+with DESI 1% survey have S/N≪ 20𝜎, therefore the impacts from
+such biases are still within the error budget. The assumed systematics
+can enlarge the large(small)-scale uncertainties from ∼ 17%(∼ 10%)
+to ∼ 18%(∼ 12%).
+We further estimate the requirements on the DECaLS calibra-
+tions for precision cosmology. We evaluate the fractional bias in the
+measured correlation function 𝑤gG, considering some residual multi-
+plicative bias 𝑚 and redshift bias Δ𝑧, and present the results in Fig. 8.
+To safely use the ∼ 50𝜎 data from the large-scale of DECaLS×BGS,
+the residual multiplicative bias alone need to be controlled within
+|𝑚| < 0.02, and the mean of the redshift distribution of the source
+galaxies ⟨𝑧⟩ need to be controlled within |Δ𝑧| < 0.03 on its own. The
+net bias considering both 𝑚 and Δ𝑧 should be controlled in between
+the orange dotted curves in Fig. 8. To safely use the cosmological
+information in both the large-scale and the small-scale, with over-
+all S/N∼ 100𝜎, we require the calibrations to have |𝑚| < 0.01 and
+|Δ𝑧| < 0.015 individually, while the net bias considering both 𝑚 and
+Δ𝑧 should be controlled in between the blue dashed curves in Fig. 8.
+We note that using tomography and combining g-g lensing mea-
+surements from different density tracers (BGS, LRGs, ELGs, and
+possibly QSOs in the future) can bring stronger S/N, so the require-
+ments on the calibration terms will be more strict. However, these
+studies will require a much larger covariance, thus more jackknife
+sub-regions and much larger overlapped regions, which are beyond
+the ability of the current data size. We leave this study to future
+works.
+4.3 Shear-ratio
+Shear-ratio is a powerful tool to probe cosmology or test systematics
+(Sánchez et al. 2021; Giblin et al. 2021), and it is insensitive to many
+small-scale physics. As shown in Table 1, DECaLS×DESI, especially
+for the BGS and LRGs, can offer very high S/N measurements at
+the small-scale. We take the BGS from the DESI 1% survey as an
+example to study this topic.
+The galaxy samples are distributed similarly to the BGS×DECaLS
+𝑛(𝑧) as in Fig. 1, but in addition, the source galaxies are further split
+into two groups: 0.6 < 𝑧𝑝 < 0.9, and 0.9 < 𝑧𝑝 < 1.5. We calculated
+the corresponding correlations 𝑤gG
+1
+and 𝑤gG
+2 , and their ratio with
+𝑅 = 𝑤gG
+2 /𝑤gG
+1 , following Eq. (4), (5) and the description in Sec. 2.2.
+The shear-ratio results are shown in Fig. 9. Following the same
+angular binning as in Fig. 3 for the correlation calculations, we use
+the two small-scale angular bins with 𝜃 <∼ 5 arcmin, since the
+three large-scale bins are expected in the direct 2-point cosmology
+study, as described in Sec. 4.1. The current small-scale information
+MNRAS 000, 1–14 (2015)
+
+0.02
+1.03
+1.02
+0.01 -
+1.01
+residual Az
+-0.0-
+1.00
+0.99
+-0.01-
+0.98
+0.97
+-0.02
+-0.02
+-0.01
+0.0
+0.01
+0.02
+residual mD&D 1%
+9
+Figure 9. The MCMC posterior PDF of the shear-ratio measurements for
+BGS×DECaLS using Eq. (4) and (5). The galaxies are distributed as in Fig. 1,
+with source galaxies split into 0.6 < 𝑧𝑝 < 0.9 and 0.9 < 𝑧𝑝 < 1.5. The
+constraint on the shear-ratio uses the two small-scale angular bins (𝜃 <∼ 5
+arcmin) as in Fig. 3. The resulting 𝑅 = 1.21+0.42
+−0.35 agrees with the theoretical
+prediction between 1.13 and 1.18. When re-scaling the covariance to the
+final overlap of DESI×DECaLS, the shear-ratio can be constrained as good
+as 𝜎𝑅 ∼ 0.04 when using the small-scale information, and 𝜎𝑅 ∼ 0.03 when
+using the full-scale.
+can constrain shear-ratio at 𝑅 = 1.21+0.42
+−0.35, which is consistent with
+our theoretical prediction (using 𝑅 = 𝑤gG
+2 /𝑤gG
+1 , Eq. (1) and (3))
+between 1.13 and 1.18. This small angular variation is due to the
+angular dependence in 𝑃(𝑘 = ℓ+1/2
+𝜒
+, 𝑧) in Eq. (1), which is not fully
+canceled when taking the ratio using correlation functions. We note
+this weak angular dependence is small and can be easily taken into
+account in the theoretical predictions.
+To predict the constraining power when full DESI finishes, we
+rescaled the covariance based on the overlapped area as in Table 1,
+and find the shear-ratio can be constrained at 𝜎𝑅 = 0.04 with the
+small-scale information, which is not used in getting the 𝑆8 con-
+straint. Considering full information for the shear-ratio study, we can
+obtain 𝜎𝑅 = 0.03. These statistical errors are comparable with the
+shear-ratio studies in (Sánchez et al. 2021) with DES-Y3 data, show-
+ing a promising future in using shear-ratio to improve cosmological
+constraint and/or to further constrain the systematics (Giblin et al.
+2021).
+4.4 Cosmic magnification
+We discussed that the ELG×DECaLS results have low S/N in Fig. 6,
+7 and Table 1, as the ELGs are mainly distributed at large-𝑧, while
+the advantage of DECaLS is at low-𝑧. On the other hand, this opens
+a window to the study of cosmic magnification by putting the ELGs
+at high-z and using shear from low-z DECaLS galaxies. We follow
+the methodology in Liu et al. (2021) and use galaxy samples dis-
+tributed as in Fig. 10. The DECaLS galaxies are located at a much
+lower photo-𝑧 compared with the ELGs, as in the targeted shear-
+magnification correlation, the shear-density correlation exists as a
+source of systematics when even a small fraction of shear galaxies
+appear at higher-𝑧 than the ELGs.
+The measurements are shown in Fig. 11. We find positive sig-
+nals at the small-scale, and null detections at the large-scale, for
+all DECaLS, KiDS, and HSC. We tested the 45-deg rotation of the
+shear, resulting in consistency with 0 on all scales for all the source
+samples. Considering the similar calculation with eBOSS ELGs 3
+and DECaLS sources as a reference, we found the measurements
+are consistent with 0 on all scales, see Appendix B for details. In
+the measurements of Fig. 11, the null detections at the large-scale
+could be due to cosmic variance or some negative systematics such
+as intrinsic alignment. The positive measurements at the small-scale
+could be due to the targeted magnification signals, the cosmic vari-
+ance, or photo-𝑧 errors. We note to separate these different signals,
+either a stronger signal with clear angular dependencies or additional
+observables are needed to break the degeneracy.
+As a further step, we present an effective amplitude fitting of 𝑔𝜇,eff
+for the magnification signals, following Eq. (7), in Table 2. We find
+∼ 1𝜎 measurement for KiDS and ∼ 2𝜎 measurement for DECaLS
+and HSC. Considering the ELG samples are quite similar as shown
+in Fig. 10, and the three best-fit 𝑔����,eff-amplitudes are consistent, we
+evaluated the combined best-fit, achieving ∼ 3𝜎 significance. The
+covariance between different surveys is ignored for the combined
+estimation, as shot noise is more dominant in this case than the cosmic
+variance. Additionally, we find that by including shear galaxies from
+0 < 𝑧𝑝 < 0.4, the significance of magnification detection drops, due
+to the low-z data having much weaker lensing efficiency as in Eq. (2),
+and is mainly contributing noise.
+The fitting goodness of the reduced-𝜒2 (defined by the 𝜒2 between
+the best-fit and the data, divided by the degree of freedom) is gener-
+ally close to ∼ 1 for each case. This shows no significant deviation
+between the model and the data. The detected ∼ 3𝜎 positive signal
+can be either due to the cosmic magnification, or very similar stochas-
+tic photo-z outliers between the three lensing surveys. As DECaLS,
+KiDS and HSC have totally different photometric bands, photo-z al-
+gorithms, and training samples, we think the detected signals are less
+likely due to the similar photo-z outliers, and more likely to be the
+cosmic magnification signal. Therefore, by assuming the combined
+best-fit of 𝑔𝜇,eff ∼ 6.1 as the true value and rescaling the covariance
+similar to Table 1, we expect ∼ 10𝜎 detection for DECaLS DR9,
+which is very promising for a stage III lensing survey. By then, with
+a better understanding of the systematics such as IA and photo-𝑧 out-
+lier, these cross-correlations can bring very promising constraining
+power in studying cosmic magnification. We can choose to: (1) cut a
+complete and flux-limited sample and compare it with the flux func-
+tion; (2) try to use the given DESI completeness and flux function to
+find a relation of 𝑔𝜇,eff(𝛼) rather than 𝑔𝜇 = 2(𝛼 − 1); (3) compare
+with realistic mocks to infer 𝑔𝜇,eff; (4) add an artificial lensing sig-
+nal 𝜅 to real data and infer 𝑔𝜇,eff as a response 𝜕𝛿Lg /𝜕𝜅, similar to
+MetaCalibration (Sheldon & Huff 2017; Huff & Mandelbaum 2017).
+5 CONCLUSIONS
+In this work, we study the cross-correlations between DESI 1% sur-
+vey galaxies and shear measured from DECaLS, one of the imaging
+surveys for DESI target selection. For the 1% DESI data, DECaLS
+can have comparable performances compared with the main stage-III
+lensing surveys KiDS and HSC. More specifically, we measure the
+cross-correlations of DESI BGS/LRGs/ELGs × different shear cat-
+alog, shown in Fig. 2, 5 and 7. We forecast the level of significance
+with full DESI data in Table 1. Assuming systematic errors can be
+cleaned with high precision in the future, we find the large-scale S/N
+could reach > 50𝜎 for DECaLS×BGS, > 15𝜎 for DECaLS×LRG,
+3 https://www.sdss.org/surveys/eboss/
+MNRAS 000, 1–14 (2015)
+
+1.0
+0.8
+0.6
+P
+0.4
+0.2
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+R (1% survey)10
+J. Yao et al.
+Figure 10. The redshift distribution for high-z ELGs (1 < 𝑧 < 1.6) and low-z
+source galaxies (0.4 < 𝑧𝑝 < 0.7) for magnification study. The choice of such
+a large redshift gap is to prevent potential leakage due to photo-𝑧 inaccuracy.
+The numbers in the labels are the number of galaxies in the overlapped region.
+Table 2. This table shows the best-fit amplitude 𝑔𝜇,eff for the cosmic mag-
+nification. The upper part corresponds to the results in Fig. 11 for DECaLS,
+KiDS, HSC, and the combination of them (the “all” case). We find with the
+DESI 1% survey, we can already detect cosmic magnification at ∼ 3.1𝜎 for
+the shear galaxies distributed at 0.4 < 𝑧𝑝 < 0.7, while the 𝑧𝑝 < 0.4 galaxies
+are mainly contributing noise as it corresponding lensing efficiency (Eq. (2))
+is low. The degree of freedom is calculated as 𝑑𝑜 𝑓 = 𝑁data − 𝑁para. We see
+no significant deviation between data and model as 𝜒2/𝑑𝑜 𝑓 ∼ 1.
+Case
+𝑔𝜇,eff
+S/N
+𝜒2/𝑑𝑜 𝑓
+DECaLS 0.4 < 𝑧𝑝 < 0.7
+10.6+5.2
+−5.8
+1.8𝜎
+0.6/1
+KiDS 0.4 < 𝑧𝑝 < 0.7
+4.2+6.0
+−5.7
+0.7𝜎
+1.3/1
+HSC 0.4 < 𝑧𝑝 < 0.7
+5.6+2.3
+−2.3
+2.4𝜎
+1.1/1
+all 0.4 < 𝑧𝑝 < 0.7
+6.1+1.9
+−2.0
+3.1𝜎
+3.9/5
+all 0 < 𝑧𝑝 < 0.7
+5.3+2.0
+−2.0
+2.7𝜎
+12.5/11
+and > 10𝜎 for DECaLS×ELG, which are very promising before the
+stage IV surveys come out.
+We point out that the main difficulty in obtaining DECaLS cos-
+mology is the calibrations for the systematics. In order to safely use
+the large-scale ∼ 50𝜎 information of BGS×DECaLS, we need to
+achieve the minimum requirements on: (1) the multiplicative bias of
+|𝑚| < 0.02 and (2) the mean of redshift distribution |Δ𝑧| < 0.03. To
+safely use the full-scale ∼ 100𝜎 data, we required |𝑚| < 0.01 and
+|Δ𝑧| < 0.015 for future calibrations. The requirement could be even
+higher when combining different observables, but it will require a
+larger footprint than the 1% survey for the study. These requirements
+are essential guides for future calibrations and studies on cosmology.
+Figure 11. The magnification(ELGs)-shear correlation measurements, cor-
+responding to the galaxy samples in Fig. 10. The theoretical curves are based
+on Eq. (6), assuming 𝑔𝜇,eff = 1 as a reference. The {small-scale, large-scale}
+detection significance for ELG×DECaLS are {2.2, 0.3}, for ELG×KiDS are
+{1.2, -0.3}, and for ELG×HSC are {2.8, -0.3}. The negative values at the
+large-scale represent negative measurements, which might be due to shot
+noise, sample variance, or impact from systematics with negative values, like
+intrinsic alignment if there exists some photo-z outlier.
+To fully use the advantage of DECaLS, we further explored two
+promising observables, the shear-ratio, and the cosmic magnification.
+We show the current 1% BGS data can constrain shear-ratio with
+𝜎𝑅 ∼ 0.4, while the full DESI BGS can give 𝜎𝑅 ∼ 0.04 using only
+the small-scale information, as shown in Fig. 9. Furthermore, weak
+detections of potential cosmic magnification are shown in Fig. 11
+and Table 2. We discussed how the possible systematics can affect
+this signal in Sec. 4.4. We also expect DECaLS to have a strong
+contribution (∼ 10𝜎 detection) to future magnification studies, if the
+observed signals in this work are not due to fluctuations.
+To summarize, DECaLS lensing is a very promising tool that can
+enrich the cosmological output of DESI. It will bring new cosmolog-
+ical information with its huge footprint. It has great advantages in the
+large-scale and the low-𝑧 information, after carefully addressing the
+systematics. It will offer strong S/N for shear-ratio study, and good
+potential in measuring cosmic magnification. Careful calibrations
+of the shear and redshift distribution can result in very promising
+outcomes.
+ACKNOWLEDGEMENTS
+We thank Xiangkun Liu, Weiwei Xu, and Jun Zhang for their helpful
+discussions. We thank Chris Blake, Daniel Gruen, and Benjamin
+Joachimi for their contribution during the DESI collaboration-wide
+review.
+HYS acknowledges the support from NSFC of China under grant
+11973070, the Shanghai Committee of Science and Technology grant
+No.19ZR1466600 and Key Research Program of Frontier Sciences,
+CAS, Grant No. ZDBS-LY-7013. PZ acknowledges the support of
+NSFC No. 11621303, the National Key R&D Program of China
+MNRAS 000, 1–14 (2015)
+
+ELG 86887
+4
+DECaLS 248178
+3
+N
+n2
+1
+0
+ELG 45516
+4 
+KiDS 520445
+3
+N
+n
+2
+1
+0
+4
+ELG 39950
+HSC 988384
+3 
+N
+m2
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Z or Zp10-4
+ELG DECaLS
+10-5
+ELG KiDS
+ELG HSC
+ELG DECaLS
+ELG KiDS
+10-6
+ELG HSC
+data / model
+20
+100
+101
+θ [arcmin]D&D 1%
+11
+2020YFC22016. JY acknowledges the support from NSFC Grant
+No.12203084, the China Postdoctoral Science Foundation Grant No.
+2021T140451, and the Shanghai Post-doctoral Excellence Program
+Grant No. 2021419. We acknowledge the support from the sci-
+ence research grants from the China Manned Space Project with
+NO. CMS-CSST-2021-A01, CMS-CSST-2021-A02 and NO. CMS-
+CSST-2021-B01.
+We acknowledge the usage of the following packages pyccl4,
+treecorr5, healpy6, matplotlib7, emcee8, corner9, astropy10, pan-
+das11, scipy12, dsigma13 for their accurate and fast performance
+and all their contributed authors.
+This research is supported by the Director, Office of Science,
+Office of High Energy Physics of the U.S. Department of Energy
+under Contract No. DE–AC02–05CH11231, and by the National
+Energy Research Scientific Computing Center, a DOE Office of Sci-
+ence User Facility under the same contract; additional support for
+DESI is provided by the U.S. National Science Foundation, Divi-
+sion of Astronomical Sciences under Contract No. AST-0950945 to
+the NSF’s National Optical-Infrared Astronomy Research Labora-
+tory; the Science and Technologies Facilities Council of the United
+Kingdom; the Gordon and Betty Moore Foundation; the Heising-
+Simons Foundation; the French Alternative Energies and Atomic
+Energy Commission (CEA); the National Council of Science and
+Technology of Mexico (CONACYT); the Ministry of Science and In-
+novation of Spain (MICINN), and by the DESI Member Institutions:
+https://www.desi.lbl.gov/collaborating-institutions.
+The DESI Legacy Imaging Surveys consist of three individual
+and complementary projects: the Dark Energy Camera Legacy Sur-
+vey (DECaLS), the Beijing-Arizona Sky Survey (BASS), and the
+Mayall z-band Legacy Survey (MzLS). DECaLS, BASS and MzLS
+together include data obtained, respectively, at the Blanco telescope,
+Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok
+telescope, Steward Observatory, University of Arizona; and the May-
+all telescope, Kitt Peak National Observatory, NOIRLab. NOIRLab
+is operated by the Association of Universities for Research in As-
+tronomy (AURA) under a cooperative agreement with the National
+Science Foundation. Pipeline processing and analyses of the data
+were supported by NOIRLab and the Lawrence Berkeley National
+Laboratory. Legacy Surveys also uses data products from the Near-
+Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a
+project of the Jet Propulsion Laboratory/California Institute of Tech-
+nology, funded by the National Aeronautics and Space Adminis-
+tration. Legacy Surveys was supported by: the Director, Office of
+Science, Office of High Energy Physics of the U.S. Department of
+Energy; the National Energy Research Scientific Computing Center,
+a DOE Office of Science User Facility; the U.S. National Science
+Foundation, Division of Astronomical Sciences; the National Astro-
+nomical Observatories of China, the Chinese Academy of Sciences
+and the Chinese National Natural Science Foundation. LBNL is man-
+4 https://github.com/LSSTDESC/CCL, (Chisari et al. 2019)
+5 https://github.com/rmjarvis/TreeCorr, (Jarvis et al. 2004)
+6 https://github.com/healpy/healpy, (Górski et al. 2005; Zonca et al.
+2019)
+7 https://github.com/matplotlib/matplotlib, (Hunter 2007)
+8 https://github.com/dfm/emcee, (Foreman-Mackey et al. 2013)
+9 https://github.com/dfm/corner.py, (Foreman-Mackey 2016)
+10 https://github.com/astropy/astropy,
+(Astropy
+Collaboration
+et al. 2013)
+11 https://github.com/pandas-dev/pandas
+12 https://github.com/scipy/scipy, (Jones et al. 01 )
+13 https://github.com/johannesulf/dsigma
+aged by the Regents of the University of California under contract to
+the U.S. Department of Energy. The complete acknowledgments can
+be found at https://www.legacysurvey.org/.
+The authors are honored to be permitted to conduct scientific
+research on Iolkam Du’ag (Kitt Peak), a mountain with particular
+significance to the Tohono O’odham Nation.
+DATA AVAILABILITY
+The data used to produce the figures in this work are available through
+https://doi.org/10.5281/zenodo.7322710 following DESI
+Data Management Plan.
+The inclusion of a Data Availability Statement is a requirement for
+articles published in MNRAS. Data Availability Statements provide
+a standardized format for readers to understand the availability of
+data underlying the research results described in the article. The
+statement may refer to original data generated in the course of the
+study or to third-party data analyzed in the article. The statement
+should describe and provide means of access, where possible, by
+linking to the data or providing the required accession numbers for
+the relevant databases or DOIs.
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+APPENDIX A: THEORETICAL COVARIANCE
+We test the Gaussian covariance assumption being used in Table 1 in
+this section. We use DECaLS×BGS and KiDS×BGS as examples,
+using the same galaxy number densities and redshift distributions as
+in Fig. 1, and the same area as shown in Table 1. The angular power
+spectrum 𝐶gG(ℓ) is calculated within range 10 < ℓ < 10000, binned
+with Δℓ = 0.2ℓ, thus total 37 angular bins. We follow the procedures
+in Joachimi et al. (2021) and divide the components into Gaussian
+covariance, connected non-Gaussian covariance, and super-sample
+covariance.
+MNRAS 000, 1–14 (2015)
+
+D&D 1%
+13
+Figure A1. The theoretical covariance matrix (normalized, i.e. correlation
+coefficient) for the DECaLS×BGS angular power spectrum, corresponding
+to the measurements in Fig. 3 and the DECaLS results in Fig. 2. It is clear the
+Gaussian component in the total covariance is much larger than the connected
+non-Gaussian component and the super-sample covariance component.
+Figure A2. The theoretical covariance matrix (normalized, i.e. correlation
+coefficient) for the KiDS×BGS angular power spectrum, corresponding to
+the measurements of the KiDS results in Fig. 2. The Gaussian component in
+the total covariance is still the dominant part. But the connected non-Gaussian
+component and the super-sample covariance component are relatively larger
+than Fig. A1 and are no longer negligible.
+The Gaussian covariance is calculated by
+CovG(ℓ1, ℓ2) =
+𝛿ℓ1,ℓ2
+(2ℓ + 1)Δℓ 𝑓sky
+�
+(𝐶gG)2 + (𝐶gg + 𝑁gg)(𝐶GG + 𝑁GG)
+�
+,
+(A1)
+where 𝛿ℓ1,ℓ2 is the Kronecker delta function; 𝐶gG, 𝐶gg and
+𝐶GG are the galaxy-lensing, galaxy-galaxy, lensing-lensing angu-
+lar power spectrum, respectively; 𝑁gg = 4𝜋 𝑓sky/𝑁g and 𝑁GG =
+4𝜋 𝑓sky𝛾2rms/𝑁G are the shot noise for 𝐶gg and 𝐶GG, where 𝑓sky is
+the fraction of sky of the overlapped area, 𝑁g and 𝑁G are the total
+number of the galaxies for the lens and source.
+The connected non-Gaussian covariance (Takada & Jain 2004) is
+calculated by
+CovcNG(ℓ1, ℓ2) =
+∫
+𝑑𝜒
+𝑏2g𝑛2
+l (𝜒)𝑞2s (𝜒)
+𝜒6
+𝑇m
+�ℓ1 + 1/2
+𝜒
+, ℓ2 + 1/2
+𝜒
+, 𝑎(𝜒)
+�
+,
+(A2)
+where 𝑛l and 𝑞s are the lens distribution and source lensing efficiency,
+𝑏g denotes the lens galaxy bias, 𝜒 denotes the comoving distance,
+same as those in Eq. (1); 𝑇m is the matter trispectrum, calculated
+using a halo model formalism (Joachimi et al. 2021). We assume the
+NFW halo profile (Navarro et al. 1996) with a concentration-mass
+relation (Duffy et al. 2008), a halo mass function (Tinker et al. 2008)
+and a halo bias (Tinker et al. 2010).
+The super-sample covariance (Takada & Hu 2013) is calculated
+by
+CovSSC(ℓ1, ℓ2) =
+∫
+𝑑𝜒
+𝑏2g𝑛2
+l (𝜒)𝑞2s (𝜒)
+𝜒6
+𝜕𝑃𝛿(ℓ1/𝜒)
+𝜕𝛿b
+𝜕𝑃𝛿(ℓ2/𝜒)
+𝜕𝛿b
+𝜎2
+b (𝜒),
+(A3)
+where the derivative of 𝜕𝑃𝛿/𝜕𝛿b gives the response of the matter
+power spectrum to a change of the background density contrast 𝛿b,
+while 𝜎2
+b denote the variance of the background matter fluctuations
+in the given footprint. In this test, we use a circular disk that covers
+the same area as the given survey to calculate 𝜎2
+b .
+The calculation is performed with the halo model tools in pyccl.
+We show the results of DECaLS×BGS in Fig. A1 and KiDS×BGS in
+Fig. A2. It is clear that the contribution from connected non-Gaussian
+covariance and super-sample covariance in DECaLS is negligible, so
+a Gaussian covariance can be fairly assumed for DECaLS in Table 1.
+The Gaussian covariance is still dominant in KiDS, however, the
+contribution from the other two is not negligible. Therefore, due to
+the small footprint, the forecasted S/N for KiDS and HSC in Table 1
+no longer scales exactly with the overlapped area.
+We note that this test for different components of the covariance
+is only used to make an estimated comparison. Before using those
+covariances directly in the study, one needs to take care of the non-
+linear galaxy bias 𝑏g, the exact shape of the footprint that produces
+𝜎2
+b , and build simulations to validate the accuracy of the theoretical
+covariance transferring from angular power spectrum to correlation
+functions as in Joachimi et al. (2021). Therefore, we choose to stick
+with the data-driven jackknife covariance introduced in the main text,
+while we note that this effect could potentially reduce the forecasted
+S/N for KiDS and HSC in Table 1.
+APPENDIX B: EBOSS ELGS × DECALS SHEAR
+We show the cosmic magnification measurements using eBOSS
+ELGs × DECaLS shear, following a similar procedure as described
+in Sec. 2.3 and 4.4. The overlapped area between eBOSS ELGs and
+DECaLS shear is ∼ 930 deg2, which enables us to use 200 jackknife
+subregions and 5 angular bins, while we calculate the correlation in
+the angular range of 0.5 < 𝜃 < 120 arcmin, which is wider than
+Fig. 3, see discussions in Sec 4.1.
+In Fig. B1 we show the galaxy redshift distribution being used
+in this measurement. We see that the eBOSS ELGs are distributed
+at lower redshift compared with DESI ELGs in Fig. 10, and more
+galaxies are used in this eBOSS measurement. The corresponding
+MNRAS 000, 1–14 (2015)
+
+1.0
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+total cov
+Gaussian cov
+0.020
+0.06
+0.015
+0.04
+0.010
+0.005
+0.02
+connected non-Gaussian cov
+super-sample cov1.0
+0.8
+0.8
+0.6
+0.6
+0.4
+0.4
+0.2
+0.2
+0.0
+total cov
+Gaussian cov
+0.10
+0.3
+0.08
+0.06
+0.2
+0.04
+0.1
+0.02
+connected non-Gaussian cov
+super-sample cov14
+J. Yao et al.
+Figure B1. The galaxy redshift distribution for the eBOSS ELGs (blue) and
+photo-z distribution for DECaLS (orange). We use 0 < 𝑧𝑝 < 0.5 for DECaLS
+and 𝑧 > 0.7 for eBOSS ELGs. The redshift ranges are generally lower than
+Fig. 10 as eBOSS ELGs are at lower redshift than DESI ELGs.
+Figure B2. The magnification(ELGs)-shear correlation measurements for
+eBOSS×DECaLS. Unlike Fig. 11 for DESI, this measurement is consistent
+with 0.
+correlation function measurement is shown in Fig. B2, which is con-
+sistent with 0. We think this is due to the fact that the galaxy number
+density for the eBOSS ELGs is much lower than the DESI ELGs,
+leading to a larger shot noise.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–14 (2015)
+
+eBOSS ELG 163690
+4
+DECaLS2597924
+3
+N
+n
+2
+1
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+z or
+Zp0.0003
+theory gμ= 1
+theory gμ= - 3.1
+0.0002
+6.0
+0.0001
+0.0000
+-0.0001
+-0.0002
+-0.0003
+100
+101
+102
+ [arcmin]

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+arXiv:2301.13433v1  [math.AP]  31 Jan 2023
+ON WELL-POSEDNESS RESULTS FOR THE CUBIC-QUINTIC NLS ON T3
+YONGMING LUO, XUEYING YU, HAITIAN YUE AND ZEHUA ZHAO
+Abstract. We consider the periodic cubic-quintic nonlinear Schr¨odinger equation
+(i∂t + ∆)u = µ1|u|2u + µ2|u|4u
+(CQNLS)
+on the three-dimensional torus T3 with µ1, µ2 ∈ R \ {0}.
+As a first result, we establish the small
+data well-posedness of (CQNLS) for arbitrarily given µ1 and µ2. By adapting the crucial perturbation
+arguments in [33] to the periodic setting, we also prove that (CQNLS) is always globally well-posed in
+H1(T3) in the case µ2 > 0.
+Contents
+1.
+Introduction and main results
+1
+2.
+Preliminaries
+3
+3.
+Proof of Theorem 1.1
+7
+4.
+Proof of Theorem 1.2
+8
+References
+10
+1. Introduction and main results
+In this paper, we study the cubic-quintic nonlinear Schr¨odinger equation (CQNLS)
+(1.1)
+(i∂t + ∆x)u = µ1|u|2u + µ2|u|4u
+on the three-dimensional torus T3, where µ1, µ2 ∈ R \ {0} and T = R/2πZ. The CQNLS (1.1) arises
+in numerous physical applications such as nonlinear optics and Bose-Einstein condensate. Physically,
+the nonlinear potentials |u|2u and |u|4u model the two- and three-body interactions respectively and the
+positivity or negativity of µ1 and µ2 indicates whether the underlying nonlinear potential is repulsive
+(defocusing) or attractive (focusing). We refer to, for instance, [10, 11, 27] and the references therein for
+a more comprehensive introduction on the physical background of the CQNLS (1.1). Mathematically,
+the CQNLS model (1.1) on Euclidean spaces Rd (d ≤ 3) has been intensively studied in [3, 4, 6, 18, 19,
+22, 23, 25, 26, 28, 33], where well-posedness and long time behavior results for solutions of (1.1) as well
+as results for existence and (in-)stability of soliton solutions of (1.1) were well established.
+In this paper, we aim to give some first well-posedness results for (1.1) in the periodic setting, which, to
+the best of our knowledge, have not existed to that date. We also restrict ourselves to the most appealing
+case d = 3, where the quintic potential is energy-critical. (By ‘energy-critical’, we mean the energy of
+solution is invariant under the scaling variance. See [9] for more details.) In this case, the well-posedness
+of (1.1) shall also depend on the profile of the initial data and the analysis becomes more delicate and
+challenging.
+Our first result deals with the small data well-posedness of (1.1), which is given in terms of the function
+spaces Z′(I), X1(I) defined in Section 2 for a given time slot I.
+2020 Mathematics Subject Classification. Primary: 35Q55; Secondary: 35R01, 37K06, 37L50.
+Key words and phrases. Nonlinear Schr¨odinger equation, global well-posedness, perturbation theory.
+1
+
+2
+YONGMING LUO, XUEYING YU, HAITIAN YUE AND ZEHUA ZHAO
+Theorem 1.1 (Small data well-posedness). Consider (1.1) on a time slot I = (−T, T ) ⊂ R with some
+T ∈ (0, ∞). Let u0 ∈ H1(T3) satisfies ∥u0∥H1(T3) ≤ E for some E > 0. Then there exists δ = δ(E, T ) > 0
+such that if
+∥eit∆u0∥Z′(I) ≤ δ,
+(1.2)
+then (1.1) admits a unique strong solution u ∈ X1(I) with initial data u(0, x) = u0(x).
+The proof of Theorem 1.1 is based on a standard application of the contraction principle. Nonetheless,
+one of the major challenges in proving well-posedness of dispersive equations on tori is the rather exotic
+Strichartz estimates, leading in most cases to very technical and cumbersome proofs. In the energy-
+subcritical setting, Strichartz estimates for periodic nonlinear Schr¨odinger equations (NLS) were first
+established by Bourgain [1] by appealing to the number-theoretical methods.
+In our case, where an
+energy-critical potential is present, we shall make use of the Strichartz estimates introduced by Herr-
+Tataru-Tzvetkov [14] based on the atomic space theory, which in turn initiates applications of the function
+spaces defined in Section 2. Notice also that in comparison to the purely quintic NLS model studied in
+[14], an additional cubic term should also be dealt in our case.
+A new bilinear estimate on T3 will
+therefore be proved in order to obtain a proper estimate for the cubic potential, and we refer to Lemma
+3.2 for details. For interested readers, we also refer to [7, 8, 14, 15, 16, 17, 29, 30, 32, 34, 35, 36] for
+further well-posedness results for NLS (with single nonlinear potential) on tori or waveguide manifolds
+based on the atomic space theory. (See [24, 31] for other dispersive equations on waveguides.)
+Despite that small data well-posedness results are satisfactory to certain extent, it is more interesting
+(and hence also more challenging) to deduce well-posedness results where the initial data are not neces-
+sarily small. We focus here on the particular scenario where the quintic potential is repulsive (µ2 > 0),
+which is motivated by the following physical concern: Consider for instance the focusing cubic NLS1
+(i∂t + ∆)u = −|u|2u
+(1.3)
+on Rd with d ∈ {2, 3}. By invoking the celebrated Glassey’s identity [12] one may construct finite time
+blow-up solutions of (1.3) for initial data lying in weighted L2-spaces or satisfying radial symmetric
+conditions, see for instance [5] for a proof. Surprisingly, in contrast to the rigorously derived blow-up
+results, collapse of the wave function does not appear in many actual experiments. It is therefore suggested
+to incorporate a higher order repulsive potential into (1.3), the case that the repulsive potential is taken
+as the three-body interaction leads to the study of CQNLS (1.1). More interestingly, it turns out that
+in the presence of a quintic stabilizing potential, (1.1) is in fact globally well-posed for arbitrary initial
+data in H1(Rd). While for d = 2 this follows already from conservation laws and the energy-subcritical
+nature of (1.1) on R2, the proof in the case d = 3, where the quintic potential becomes energy-critical, is
+more involved. A rigorously mathematical proof for confirming such heuristics in d = 3 was first given by
+Zhang [33]. The idea from [33] can be summarized as follows: We may consider (1.1) as a perturbation
+of the three dimensional defocusing quintic NLS
+(i∂t + ∆)u = |u|4u
+(1.4)
+whose global well-posedness in ˙H1(R3) was shown in [9]. We then partition the time slot I into disjoint
+adjacent small intervals I = ∪m
+j=0 Ij. On each of these intervals, the cubic term is expected to be “small”
+because of the smallness of the subinterval, and by invoking a stability result we may prove that (1.1) is
+well-posed on a given Ij. Based on the well-posedness result on Ij we are then able to prove the same
+result for the consecutive interval Ij+1 and so on. Starting from the interval I0 and repeating the previous
+procedure inductively over all Ij follows then the desired claim.
+Inspired by the result given in [33], we aim to prove the following analogous global well-posedness result
+for (1.1) on T3 in the case µ2 > 0.
+1When d = 1, the mass-subcritical nature of the nonlinear potential, combining with conservation of mass and energy,
+guarantees the global well-posedness of (1.3) in H1(R) as well as H1(T).
+
+ON WELL-POSEDNESS RESULTS FOR CQNLS ON T3
+3
+Theorem 1.2 (Global well-posedness in the case µ2 > 0). Assume that µ2 > 0. Then (1.1) is globally
+well-posed in H1(T3) in the sense that for any T > 0 and u0 ∈ H1(T3), (1.1) possesses a solution
+u ∈ X1(I) on I = (−T, T ) with u(0) = u0.
+Remark 1.3. We note that one can also obtain the waveguide analogues of Theorem 1.2, (i.e. considering
+(1.1) posed on R2 × T and R × T2) with suitable modifications. Moreover, for the R2 × T case, scattering
+behavior is also expected according to existing literature (see [35]). However, the scattering result require
+a lot more than this GWP scheme and we leave it for future considerations.
+Remark 1.4. It is worth mentioning that the same global well-posedness result for the supercubic-quintic
+NLS
+(i∂t + ∆)u = µ1|u|p−1u + µ2|u|4u,
+for
+3 < p < 5,
+is expected to be yielded by adapting the nonlinear estimates in Section 3 into the fractional product
+case (see [21] for reference, see also [33] for the Euclidean case). We leave it for interested readers.
+We follow closely the same lines from [33] to prove Theorem 1.2. In comparison to the Euclidean case,
+there are essentially two main new ingredients needed for the proof of Theorem 1.2:
+(i) The Black-Box-Theory from [9] is replaced by the one from [16] for (1.4) on T3.
+(ii) The estimates are correspondingly modified (in a very technical and subtle way) in order to apply
+the Strichartz estimates based on the atomic space theory.
+We refer to Section 4 for the proof of Theorem 1.2 in detail. For further applications of such interesting
+perturbation arguments on NLS with combined powers, we also refer to [28].
+Remark 1.5. By a straightforward scaling argument it is not hard to see that both Theorems 1.1 and 1.2
+extend verbatim to the case where T3 is replaced by any rational torus. Such direct scaling argument,
+however, does not apply to irrational tori. Nevertheless, thanks to the ground breaking work of Bourgain
+and Demeter [2] we also know that the Strichartz estimates established in [14] are in fact available for
+irrational tori, by which we are thus able to conclude that Theorems 1.1 and 1.2 indeed remain valid for
+arbitrary tori regardless of their rationality. For simplicity we will keep working with the torus T3 in the
+rest of the paper.
+We outline the structure of the rest of the paper.
+In Section 2, we summarize the notations and
+definitions which will be used throughout the paper and define the function spaces applied in the Cauchy
+problem (1.1). In Sections 3 and 4, we prove Theorems 1.1 and 1.2 respectively.
+Acknowledgment. Y. Luo was funded by Deutsche Forschungsgemeinschaft (DFG) through the Priority
+Programme SPP-1886 (No. NE 21382-1). H. Yue was supported by the Shanghai Technology Innovation
+Action Plan (No. 22JC1402400), a Chinese overseas high-level young talents program (2022) and the
+start-up funding of ShanghaiTech University. Z. Zhao was supported by the NSF grant of China (No.
+12101046, 12271032), Chinese overseas high-level young talents program (2022) and the Beijing Institute
+of Technology Research Fund Program for Young Scholars.
+2. Preliminaries
+In this section, we first discuss notations used in the rest of the paper, introduce the function spaces
+with their properties that we will be working on, and list some useful tools from harmonic analysis.
+2.1. Notations. We use the notation A ≲ B whenever there exists some positive constant C such that
+A ≤ CB.
+Similarly we define A ≳ B and we use A ∼ B when A ≲ B ≲ A.
+For simplicity, we
+hide in most cases the dependence of the function spaces on their spatial domain in their indices. For
+example L2
+x = L2(T3), ℓ2
+k = ℓ2(Z3) and so on. However, when the space is involved with time we still
+display the underlying temporal interval such as Lp
+t,x(I), Lp
+tLq
+x(I), L∞
+t ℓ2
+k(R) etc. We also frequently write
+∥ · ∥p := ∥ · ∥Lp
+x.
+
+4
+YONGMING LUO, XUEYING YU, HAITIAN YUE AND ZEHUA ZHAO
+2.2. Fourier transforms and Littlewood-Paley projections. Throughout the paper we use the
+following Fourier transformation on T3:
+(Ff)(ξ) = �f(ξ) = (2π)− 3
+2
+�
+Td f(x)e−ix·ξ dx
+for ξ ∈ Z3. The corresponding Fourier inversion formula is then given by
+f(x) = (2π)− 3
+2 �
+ξ∈Z3
+(Ff)(ξ)eix·ξ.
+By definition, the Schr¨odinger propagator eit∆ is defined by
+�
+Feit∆f
+�
+(ξ) = e−it|ξ|2(Ff)(ξ).
+Next we define the Littlewood-Paley projectors. We fix some even decreasing function η ∈ C∞
+c (R; [0, 1])
+satisfying η(t) ≡ 1 for |t| ≤ 1 and η(t) ≡ 0 for |t| ≥ 2. For a dyadic number N ≥ 1 define ηN : Z3 → [0, 1]
+by
+ηN(ξ) = η(|ξ|/N) − η(2|ξ|/N),
+N ≥ 2,
+ηN(ξ) = η(|ξ|),
+N = 1.
+Then the Littlewood-Paley projector PN (N ≥ 1) is defined as the Fourier multiplier with symbol ηN.
+For any N ∈ (0, ∞), we also define
+P≤N :=
+�
+M∈2N,M≤N
+PM,
+P>N :=
+�
+M∈2N,M>N
+PM.
+2.3. Strichartz estimates. As already pointed out in the introductory section, unlike the Euclidean
+case, the Strichartz estimates on (rational or irrational) tori are generally proved in a highly non-trivial
+way and in most cases only frequency-localized estimates can be deduced. For our purpose we will make
+use of the following Strichartz estimate proved by Bourgain and Demeter [2] (see also [1, 20]).
+Proposition 2.1 (Frequency-localized Strichartz estimates on T3, [2]). Consider the linear Schr¨odinger
+propagator eit∆ on a (rational or irrational) three-dimensional torus. Then for p > 10
+3 we have for any
+time slot I with |I| ≤ 1
+(2.1)
+∥eit∆PNf∥Lp
+t,x(I×T3) ≲p N
+3
+2 − 5
+p ∥PNf∥L2x(T3).
+2.4. Function spaces. Next, we define the function spaces and collect some of their useful properties
+which will be used for the Cauchy problem (1.1). We begin with the definitions of U p- and V p-spaces
+introduced in [13].
+Definition 2.2 (U p-spaces). Let 1 ≤ p < ∞, H be a complex Hilbert space and Z be the set of all finite
+partitions −∞ < t0 < t1 < ... < tK ≤ ∞ of the real line. A U p-atom is a piecewise constant function
+a : R → H defined by
+a =
+K
+�
+k=1
+χ[tk−1,tk)φk−1,
+where {tk}K
+k=0 ∈ Z and {φk}K−1
+k=0 ⊂ H with �K
+k=0 ∥φk∥p
+H = 1. The space U p(R; H) is then defined as the
+space of all functions u : R → H such that u = �∞
+j=1 λjaj with U p-atoms aj and {λj} ∈ ℓ1. We also
+equip the space U p(R; H) with the norm
+∥u∥Up := inf{
+∞
+�
+j=1
+|λj| : u =
+∞
+�
+j=1
+λjaj, λj ∈ C, aj are U p-atoms}.
+
+ON WELL-POSEDNESS RESULTS FOR CQNLS ON T3
+5
+Definition 2.3 (V p-spaces). We define the space V p(R, H) as the space of all functions v : R → H such
+that
+∥v∥V p :=
+sup
+{tk}K
+k=0∈Z
+(
+K
+�
+k=1
+∥v(tk) − v(tk−1)∥p
+H)
+1
+p < +∞,
+where we use the convention v(∞) = 0. Also, we denote by V p
+rc(R, H) the closed subspace of V p(R, H)
+containing all right-continuous functions v with
+lim
+t→−∞ v(t) = 0.
+In our context we shall set the Hilbert space H to be the Sobolev space Hs
+x with s ∈ R, which will be
+the case in the remaining parts of the paper.
+Definition 2.4 (U p
+∆- and V p
+∆-spaces in [13]). For s ∈ R we let U p
+∆Hs
+x(R) resp. V p
+∆Hs
+x(R) be the spaces
+of all functions such that e−it∆u(t) is in U p(R, Hs
+x) resp. V p
+rc(R, Hs
+x), with norms
+∥u∥Up
+∆Hsx(R) = ∥e−it∆u∥Up(R,Hsx),
+∥u∥V p
+∆Hsx(R) = ∥e−it∆u∥V p(R,Hsx).
+Having defined the U p
+∆- and V p
+∆-spaces we are now ready to formulate the function spaces for studying
+the Cauchy problem (1.1). For C = [− 1
+2, 1
+2)3 ∈ R3 and z ∈ R3 let Cz = z + C be the translated unit cube
+centered at z and define the sharp projection operator PCz by
+F(PCzf)(ξ) = χCz(ξ)F(f)(ξ),
+ξ ∈ Z3,
+where χCz is the characteristic function restrained on Cz. We then define the Xs- and Y s-spaces as
+follows:
+Definition 2.5 (Xs- and Y s-spaces). For s ∈ R we define the Xs(R)- and Y s(R)-spaces through the
+norms
+∥u∥2
+Xs(R) :=
+�
+z∈Z3
+⟨z⟩2s∥PCzu∥2
+U2
+∆(R;L2x),
+∥u∥2
+Y s(R) :=
+�
+z∈Z3
+⟨z⟩2s∥PCzu∥2
+V 2
+∆(R;L2x).
+For an interval I ⊂ R we also consider the restriction spaces Xs(I), Y s(I) etc. For these spaces we
+have the following useful embedding:
+Proposition 2.6 (Embedding between the function spaces, [13]). For 2 < p < q < ∞ we have
+U 2
+∆Hs
+x ֒→ Xs ֒→ Y s ֒→ V 2
+∆Hs
+x ֒→ U p
+∆Hs
+x ֒→ U q
+∆Hs
+x ֒→ L∞Hs
+x.
+As usual, the proofs of the well-posed results rely on the contraction principle and thus a dual norm
+estimation for the Duhamel term is needed.
+In the periodic setting, the dual norm is given as the
+N s-norm, which is defined as follows:
+Definition 2.7 (N s-norm). On a time slot I we define the N s(I)-norm for s ∈ R by
+∥h∥N s(I) = ∥
+� t
+a
+ei(t−s)∆h(s) ds∥Xs(I).
+The following proposition sheds light on the duality of the spaces N 1(I) and Y −1(I).
+Proposition 2.8 (Duality of N 1(I) and Y −1(I) in [14]). The spaces N 1(I) and Y −1(I) satisfy the
+following duality inequality
+∥f∥N 1(I) ≲
+sup
+∥v∥Y −1(I)≤1
+�
+I×T3 f(t, x)v(t, x) dxdt.
+Moreover, the following estimate holds for any smooth (H1
+x-valued) function g on an interval I = [a, b]:
+∥g∥X1(I) ≲ ∥g(a)∥H1x + (
+�
+N
+∥PN(i∂t + ∆)g∥2
+L1
+tH1x(I))
+1
+2 .
+
+6
+YONGMING LUO, XUEYING YU, HAITIAN YUE AND ZEHUA ZHAO
+For our purpose we shall also need appeal to the Z-norm which is defined as follows:
+Definition 2.9 (Z-norm). For a time slot I we define the Z(I)-norm by
+∥u∥Z(I) =
+sup
+J⊂I,|J|≤1
+(
+�
+N≥1
+N 3∥PNu∥4
+L4
+t,x(J))
+1
+4 .
+As a direct consequence of the Strichartz estimates it is easy to verify that
+∥u∥Z(I) ≲ ∥u∥X1(I).
+For those readers who are familiar with NLS on the standard Euclidean space Rd, we also note that
+intuitively, the X1- and Z-norms play exactly the same roles as the norm ∥ · ∥S1 := ∥ · ∥L∞
+t H1x∩L2
+tW 1,6
+x
+and
+L10
+t,x-norm for the quintic NLS on R3 respectively. Nevertheless, the Z-norm can not be directly applied
+to prove the well-posedness results. To that end, we introduce the Z′-norm defined by
+∥u∥Z′ := ∥u∥
+1
+2
+Z∥u∥
+1
+2
+X1
+which will be more useful for the proof of Theorem 1.1.
+2.5. Conservation laws. We end this section by introducing the mass M(u) and energy E(u) associating
+to the NLS flow (1.1):
+M(u) =
+�
+T3 |u|2 dx,
+E(u) =
+�
+T3
+1
+2|∇u|2 + µ1
+4 |u|4 + µ2
+6 |u|6 dx.
+(2.2)
+It is well-known that both mass and energy are conserved over time along the NLS flow (1.1).
+As a direct application of conservation laws and H¨older’s inequality, we have the following uniform
+estimate of the kinetic energy ∥∇u∥2
+L∞
+t L2x(I×T3) =: ∥∇u∥2
+L∞
+t L2x(I) for a solution u of (1.1). (As mentioned
+in Notations, we omit the space T3 for convenience.) We include the proof below for completeness (see
+the original argument in [33, Sec. 2.2]).
+Lemma 2.10. Let u ∈ X1(I) be a solution of (1.1) with u(0) = u0. Then
+∥∇u∥2
+L∞
+t L2x(I) ≲ E(u0) + M(u0)2.
+Proof of Lemma 2.10. Recall the mass and energy defined in (2.2). If both µ1 and µ2 are positive, it is
+easy to see that for any t
+∥∇u(t)∥2
+L2x ≲ E.
+If µ1 < 0 and µ2 > 0, then we use the following inequality for some C(µ1, µ2)
+−|µ1|
+4 |u(t, x)|4 + |µ2|
+6 |u(t, x)|6 ≥ −C(µ1, µ2)|u(t, x)|2
+to conclude that for any t
+∥∇u(t)∥2
+L2x ≲ E + M 2.
+□
+
+ON WELL-POSEDNESS RESULTS FOR CQNLS ON T3
+7
+3. Proof of Theorem 1.1
+In this section we give the proof of Theorem 1.1. As the precise value of |I| = 2T has only impact
+on the numerical constants, without loss of generality, we may also assume that |I| ≤ 1 throughout this
+section.
+We begin with recording a trilinear estimate deduced in [16].
+Lemma 3.1 (Trilinear estimate, [16]). Suppose that ui = PNiu, for i = 1, 2, 3 satisfying N1 ≥ N2 ≥
+N3 ≥ 1. Then there exists some δ > 0 such that
+∥u1u2u3∥L2
+t,x(I) ≲
+�N3
+N1
++ 1
+N2
+�δ
+∥u1∥Y 0(I)∥u2∥Z′(I)∥u3∥Z′(I).
+For dealing with the cubic term, we also need the following bilinear estimate.
+Lemma 3.2 (Bilinear estimate). Suppose that ui = PNiu, for i = 1, 2 satisfying N1 ≥ N2 ≥ 1. Then
+there exists some κ > 0 such that
+∥u1u2∥L2
+t,x(I) ≲
+�N2
+N1
++ 1
+N2
+�κ
+|I|
+1
+20 ∥u1∥Y 0(I)∥u2∥Z′(I).
+Proof of Lemma 3.2. For any cube C centered at ξ ∈ Z3 of size N2, using H¨older’s inequality and the
+Strichartz estimate (2.1), we have
+∥(PCu1)u2∥L2
+t,x(I) ≲ ∥PCu1∥L4
+t,x(I)∥u2∥L4
+t,x(I) ≲ |I|
+1
+10 ∥PCu1∥
+L
+20
+3
+t,x(I)∥u2∥L4
+t,x(I)
+≲ |I|
+1
+10 ∥PCu1∥
+U
+20
+3
+∆ L2x(I)
+�
+N
+3
+4
+2 ∥u2∥L4
+t,x(I)
+�
+≲ |I|
+1
+10 ∥PCu1∥Y 0(I)
+�
+N
+3
+4
+2 ∥u2∥L4
+t,x(I)
+�
+.
+Using the orthogonality and summability properties of Y 0(I) and the definition of Z(I), the above
+estimate provides
+(3.1)
+∥u1u2∥2
+L2
+t,x(I) ≲ |I|
+1
+5 �
+C
+∥PCu1∥2
+Y 0(I)
+�
+N
+3
+4
+2 ∥u2∥L4
+t,x(I)
+�2
+≲ |I|
+1
+5 ∥u1∥2
+Y 0(I)∥u2∥2
+Z(I),
+where the sum is over all ξ ∈ N −1
+2 Z3. It remains to prove
+(3.2)
+∥u1u2∥L2
+t,x(I) ≲
+�N2
+N1
++ 1
+N2
+�κ0
+∥u1∥Y 0(I)∥u2∥Y 1(I)
+for some κ0 > 0, the desired claim follows then from interpolating (3.1) and (3.2) and the embedding
+X1 ֒→ Y 1. Again, using the orthogonality and summability properties of Y 0(I) and Strichartz estimate
+(2.1), we obtain that
+∥u1u2∥2
+L2
+t,x(I) ≲
+�
+C
+∥(PCu1)u2∥2
+L2
+t,x(I) ≲
+�
+C
+�
+N
+1
+2
+2 ∥PCu1∥U4
+∆L2
+x(I)∥u2∥U4
+∆L2
+x(I)
+�2
+≲
+�
+C
+�
+N
+− 1
+2
+2
+∥PCu1∥Y 0(I)∥u2∥Y 1(I)
+�2
+≲ N −1
+2 ∥u1∥2
+Y 0(I)∥u2∥2
+Y 1(I),
+as desired.
+□
+As a direct consequence of the multilinear estimates deduced from Lemmas 3.1 and 3.2, we immediately
+obtain the following nonlinear estimates.
+
+8
+YONGMING LUO, XUEYING YU, HAITIAN YUE AND ZEHUA ZHAO
+Lemma 3.3 (Nonlinear estimates). For uk ∈ X1(I), k = 1, 2, 3, 4, 5, the following estimates
+���
+5
+�
+i=1
+�ui
+���
+N 1(I) ≲
+�
+{i1,...i5}={1,2,3,4,5}
+∥ui1∥X1(I) ·
+�
+ik̸=i1
+∥uik∥Z′(I),
+(3.3)
+���
+3
+�
+i=1
+�ui
+���
+N 1(I) ≲ |I|
+1
+20
+�
+{i1,i2,i3}={1,2,3}
+∥ui1∥X1(I) ·
+�
+ik̸=i1
+∥uik∥Z′(I)
+(3.4)
+hold true, where �u ∈ {u, ¯u}.
+Proof of Lemma 3.3. (3.3) and (3.4) can be proved, words by words, by using the arguments from [16,
+Lem. 3.2] and [17, Lem. 3.2], respectively, which make use of Lemma 3.1 as well as Lemma 3.2. We thus
+omit the repeating arguments.
+□
+Having all the preliminaries we are in a position to prove Theorem 1.1.
+Proof of Theorem 1.1. We define the contraction mapping
+Φ(u) := eit∆u0 − iµ1
+� t
+0
+ei(t−s)∆|u|2u ds − iµ2
+� t
+0
+ei(t−s)∆|u|4u ds.
+We aim to show that by choosing δ0 sufficiently small, the mapping Φ defines a contraction on the metric
+space
+S := {u ∈ X1(I) : ∥u∥X1(I) ≤ 2CE, ∥u∥Z′(I) ≤ 2δ},
+where C ≥ 1 is some universal constant. The space S is particularly a complete metric space equipping
+with the metric ρ(u, v) := ∥u − v∥X1(I). First we show that for δ small we have Φ(S) ⊂ S. Indeed, using
+Lemma 3.3 we obtain
+∥Φ(u)∥X1(I) ≤ ∥eit∆u0∥X1(I) + C∥u∥X1(I)∥u∥2
+Z′(I) + C∥u∥X1(I)∥u∥4
+Z′(I)
+≤ C∥u0∥H1x + C(2CE)(2Cδ)2 + C(2CE)(2Cδ)4
+≤ CE(1 + (2C)3δ2 + (2C)5δ4) ≤ 2CE,
+∥Φ(u)∥Z′(I) ≤ ∥eit∆u0∥Z′(I) + C∥u∥X1(I)∥u∥2
+Z′(I) + C∥u∥X1(I)∥u∥4
+Z′(I)
+≤ δ + C(2CE)(2Cδ)2 + C(2CE)(2Cδ)4 ≤ 2δ
+by choosing δ sufficiently small.
+It is left to show that Φ is a contraction for small δ. Again, using Lemma 3.3 we obtain
+∥Φ(u) − Φ(v)∥X1(I) ≤ C(∥u∥X1(I) + ∥v∥X1(J))(∥u∥Z′(I) + ∥v∥Z′(I))∥u − v∥X1(I)
++ C(∥u∥X1(I) + ∥v∥X1(J))(∥u∥Z′(I) + ∥v∥Z′(I))3∥u − v∥X1(I)
+≤ C(4CE)(4Cδ + (4Cδ)3)∥u − v∥X1(I) ≤ 1
+2∥u − v∥X1(I)
+by choosing δ small. This completes the proof of Theorem 1.1.
+□
+4. Proof of Theorem 1.2
+In this section we prove Theorem 1.2. Again, without loss of generality, we may assume that |I| ≤ 1
+and µ2 = 1. The goal is therefore to show that (1.1) is well-posed on I without imposing the smallness
+condition (1.2). We firstly introduce the following large data Black-Box-Theory for defocusing quintic
+NLS on T3 from [16].
+
+ON WELL-POSEDNESS RESULTS FOR CQNLS ON T3
+9
+Theorem 4.1 (GWP of the defocusing quintic NLS on T3, [16]). Consider the defocusing quintic NLS
+(i∂t + ∆)v = |v|4v
+(4.1)
+on I = (−T, T ) with |I| ≤ 1. Then for any v0 ∈ H1
+x, (4.1) possesses a unique solution v ∈ X1(I) with
+v(0) = v0. Moreover, we have
+∥v∥X1(I) + ∥v∥Z(I) ≤ C(M(v0), E(v0)) < ∞.
+(4.2)
+We are now ready to prove Theorem 1.2.
+Proof of Theorem 1.2. Consider first a subinterval J = (a, b) ⊂ I and the difference NLS equation
+(i∂t + ∆)w = µ1|v + w|2(v + w) + |v + w|4(v + w) − |v|4v
+(4.3)
+on J with w(a) = 0 and v a solution of (4.1) with v(a) = u(a). The proof of Theorem 1.2 for the interval
+J follows once we are able to prove that (4.3) possesses a unique solution w ∈ X1(J). By (4.2) and
+the definition of the Z′-norm, we may partition I into disjoint consecutive intervals I = ∪m
+j=0 Ij with
+Ij = [tj, tj+1] such that
+∥v∥Z′(Ij) ≤ η
+for some to be determined small η.
+From now on we consider those Ij such that Ij ∩ J ̸= ∅ (say
+m1 ≤ j ≤ m2). Without loss of generality we may also assume that J = ∪m2
+j=m1 Ij. Suppose at the
+moment that for a given Ij, the solution w satisfies
+max{∥w∥L∞
+t H1x(Ij), ∥w∥X1(Ij)} ≤ (2C)j|J|
+1
+20
+with some universal constant C > 0.
+We consider the contraction mapping
+Γjw := ei(t−tj)∆w(tj) − i
+� t
+tj
+ei(t−s)∆(µ1|v + w|2(v + w) + |v + w|4(v + w) − |v|4v)(s) ds
+on the set
+Sj := {w ∈ X1(Ij) : max{∥w∥L∞
+t H1
+x(Ij), ∥w∥X1(Ij)} ≤ (2C)j|J|
+1
+20 },
+which is a complete metric space with respect to the metric
+ρ(u1, u2) := ∥u1 − u2∥X1(Ij).
+We show that by choosing η and |J| small, the mapping Γj defines a contraction on Sj. Using Strichartz
+estimates, Lemma 3.3, the embedding X1 ֒→ Z′ and the inductive hypothesis
+∥w(tj)∥H1
+x ≤ (2C)j−1|J|
+1
+20
+we obtain
+max{∥Γjw∥L∞
+t H1x(Ij), ∥Γjw∥X1(Ij)}
+≤ C∥w(tj)∥H1x + �C
+4
+�
+i=1
+(∥w∥5−i
+X1(Ij)∥v∥i
+Z′(Ij) + ∥v∥X1(Ij)∥w∥5−i
+X1(Ij)∥v∥i−1
+Z′(Ij))
++ �C∥w∥5
+X1(Ij) + �C|J|
+1
+20 ∥v + w∥X1(Ij)∥v + w∥2
+Z′(Ij)
+≤
+�
+C((2C)j−1|J|
+1
+20 )
+�
++
+�
+�C
+3
+�
+i=1
+((2C)j|J|
+1
+20 )5−iηi + �C∥v∥X1(I)
+3
+�
+i=2
+((2C)j|J|
+1
+20 )5−iηi−1
++ �C∥v∥X1(I)((2C)j|J|
+1
+20 )4 + �C|J|
+1
+20 ((2C)j|J|
+1
+20 )η2
++ �C|J|
+1
+20 ∥v∥X1(I)((2C)j|J|
+1
+20 )2 + �C|J|
+1
+20 ((2C)j|J|
+1
+20 )3�
++
+�
+�C|J|
+1
+20 ∥v∥X1(I)η2 + �C((2C)j|J|
+1
+20 )η4 + �C∥v∥X1(I)((2C)j|J|
+1
+20 )η3�
+=: A1 + A2 + A3
+
+10
+YONGMING LUO, XUEYING YU, HAITIAN YUE AND ZEHUA ZHAO
+for some �C > 0. We have A1 = 1
+2(2C)j|J|
+1
+20 . By choosing η = η(∥v∥X1(I)) = η(∥u(a)∥H1x) sufficiently
+small depending on ∥u(a)∥H1x we have A3 ≤
+1
+4(2C)j|J|
+1
+20 .
+For A2, we may choose |J| ≤ �η with �η
+depending on 0 ≤ j ≤ m so that A2 ≤ 1
+4(2C)j|J|
+1
+20 is valid for all j. Indeed, the dependence of J on
+j can be expressed as on ∥u(a)∥H1x via j ≤ m ≤ C(∥v∥Z′(I)) = C(∥u(a)∥H1x), where the last equality is
+deduced from Theorem 4.1. Similarly we are able to show that by shrinking η and �η if necessary, we have
+∥Γj(u1) − Γj(u2)∥X1(Ij) ≤ 1
+2∥u1 − u2∥X1(Ij)
+for all 0 ≤ j ≤ m − 1. The proof is analogous and we hence omit the details here. The claim then follows
+from the Banach fixed point theorem.
+Now we close our proof by removing the smallness of |J|. By Lemma 2.10 we have ∥u∥L∞
+t H1x(I) < ∞.
+Thus we may choose (η, �η) = (η, �η)(∥u∥L∞
+t H1x(I)) in a way such that the previous proof is valid for
+all J = [a, b] for any a ∈ I with |J| ≤ �η. We now partition I into disjoint consecutive subintervals
+I = ∪n
+j=0 Jj with |Jj| ≤ �η for all 0 ≤ j ≤ n, and the proof follows by applying the previous step to each
+Jj and summing up.
+□
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+uide manifolds. arXiv preprint arXiv:2111.09651 (2021).
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+Differential Equations 230, 2 (2006), 422–445.
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+Yongming Luo
+Institut f¨ur Wissenschaftliches Rechnen, Technische Universit¨at Dresden
+Zellescher Weg 25, 01069 Dresden, Germany.
+Email address: yongming.luo@tu-dresden.de
+Xueying Yu
+Department of Mathematics, University of Washington
+C138 Padelford Hall Box 354350, Seattle, WA 98195,
+Email address: xueyingy@uw.edu
+Haitian Yue
+Institute of Mathematical Sciences, ShanghaiTech University
+Pudong, Shanghai, China.
+Email address: yuehaitian@shanghaitech.edu.cn
+Zehua Zhao
+Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China.
+MIIT Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing, China.
+Email address: zzh@bit.edu.cn
+

B9AyT4oBgHgl3EQfR_dM/content/tmp_files/2301.00076v1.pdf.txt

@@ -0,0 +1,1093 @@
+Higgs Boson Mass Corrections at N3LO in the Top-Yukawa Sector of the Standard
+Model
+E. A. Reyes R.∗ and A. R. Fazio†
+Departamento de F´ısica, Universidad de Pamplona,
+Pamplona - Norte de Santander, Colombia and
+Departamento de F´ısica, Universidad Nacional de Colombia,
+Bogot´a, Colombia.
+The search of new physics signals in the Higgs precision measurements plays a pivotal role in the
+High-Luminosity Large Hadron Collider (HL-LHC) and the future colliders programs. The Higgs
+properties are expected to be measured with great experimental precision, implying higher-order
+perturbative computations of the electroweak parameters from the theoretical side. In particular,
+the Higgs boson mass parameter in the Standard Model runs over several tens of MeV with a cor-
+responding large theoretical uncertainty. A more stable result under the renormalization group can
+be computed from a non-zero external momentum Higgs self-energy, for which available calculations
+include three-loop corrections in the QCD sector. In this work we present an additional contribu-
+tion, by estimating the leading non-QCD three-loop corrections to the mass of the Higgs boson in
+the Top-Yukawa sector of order y6
+t . The momentum dependent Higgs self-energy is computed in the
+tadpole-free scheme for the Higgs vacuum expectation value in the Landau gauge and the explicit
+dependence upon the Higgs boson and top quark masses is shown. The obtained result is expressed
+in dimensional regularization as a superposition of a set of master integrals with coefficients that
+are free of poles in four space-time dimensions and the corrections are evaluated numerically by the
+sector decomposition method.
+I.
+INTRODUCTION
+The experiments have recently showed that high-
+precision measurements of the observables in the elec-
+troweak (EW) sector of the Standard Model (SM) are
+moving away from the theoretical expectations. In the
+past year, the Fermilab MUON g-2 collaboration [1] pub-
+lished its results concerning the muon anomalous mag-
+netic moment, showing a discrepancy between the exper-
+imental value and the SM predictions corresponding to a
+4.2σ difference. Recently, another EW observable joins
+to this list of anomalous measurements, namely the mass
+of the W-boson. The CDF collaboration [2] reported a
+new and more precise value, MW = 80433.5 ± 9.4 MeV ,
+together with the complete dataset collected by the CDF
+II detector at the Fermilab Tevatron. The current SM
+prediction evidences a tension of 7σ compared with the
+CDF measurement, suggesting the possibility to improve
+the SM calculations or to extend the SM. New and more
+precise experiments of the SM observables can help to ex-
+plain the origin of those discrepancies, but this requires
+also an improvement on the precision of the theoreti-
+cal calculations. In particular, the Higgs boson mass is
+an input parameter in the theoretical expressions for the
+above mentioned observables and an improvement of its
+theoretical uncertainties can lead to more precise pre-
+dictions to be compared with measurements at future
+accelerators. The improvement can come from the com-
+putation of the missing higher order corrections to the
+∗ eareyesro@unal.edu.co
+† arfazio@unal.edu.co
+Higgs mass which are left out due to the assumption of
+some kinematic limit or due to the truncation of the per-
+turbative expansions at some level. In the SM, the trun-
+cation is done at three-loop order. The one- and two-loop
+level corrections to the Higgs self-energy have been com-
+pletely computed [3–7] and implemented in the public
+computer codes mr [8] and SMDR [9]. In the former mr
+code the renormalized vacuum expectation value of the
+Higgs field is defined as the minimum of the tree-level
+Higgs potential.
+The corrections to the mass parame-
+ters are consequently gauge invariant due to the explicit
+insertion of the tadpole diagrams. The disadvantage of
+this approach is that the Higgs tadpoles can include neg-
+ative powers of the Higgs quartic self-coupling leading
+to very large corrections in MS schemes that deterio-
+rates the perturbative stability. On the other hand, the
+corrections included in SMDR typically leads to stable per-
+turbative predictions but suffers from gauge dependences
+since the vacuum is defined as the minimum of the Higgs
+effective potential and therefore the tadpoles are removed
+by imposing an appropriate renormalization condition. It
+would be convenient to have a gauge independent predic-
+tion with a stable perturbative behaviour, as highlighted
+in [10, 11] where the longstanding discussion about a suit-
+able prescription for tadpole contributions in EW renor-
+malization is solved at one-loop level. Additionally, the
+three-loop corrections have been evaluated in the gauge-
+less limit where the EW contributions are disregarded.
+In this computation the external momentum dependence
+of the contributions that are proportional to g4
+sy2
+t M 2
+t is
+included, where gs is the strong coupling constant, yt is
+the top quark Yukawa coupling and Mt is the top quark
+mass. There are also included the three-loop contribu-
+arXiv:2301.00076v1  [hep-ph]  31 Dec 2022
+
+2
+tions proportional to g2
+sy4
+t M 2
+t and y6
+t M 2
+t using the 1PI
+effective potential, from which the 1PI self-energies at
+vanishing external momenta can be derived. All those
+three-loop corrections are implemented in the last ver-
+sion of SMDR [12, 13]. Although these SMDR predictions
+are rather precise, they contain a renormalization scale
+dependence of several tens of MeV implying theoretical
+uncertainties larger than the expected experimental ones,
+of about 10-20 MeV, for the Higgs boson mass measure-
+ments at the HL-LHC, ILC and FCCee [14]. A more re-
+fined calculation including the missing higher order con-
+tributions is therefore required.
+In this paper we compute an additional three-loop con-
+tribution to the mass of the Higgs boson coming from
+the non-QCD Top-Yukawa sector in the gaugeless limit
+where the three-loop Higgs self-energy corrections at or-
+der y6
+t are calculated. These three-loop corrections are
+meant to be included into the prediction of the physical
+Higgs boson mass (Mh) which comes from the complex
+pole of the Higgs propagator in an on-shell scheme and
+therefore the Higgs self-energies are evaluated at non-
+vanishing external momentum, pµ ̸= 0. Since the ratio
+Mh/Mt ≈ 0.6 is not a really small expansion parame-
+ter, the leading three-loop corrections may receive sig-
+nificant contributions from the external momentum de-
+pendent terms evaluated at p2 = M 2
+h. Additionally, the
+inclusion of the non-vanishing external momentum self-
+energies are expected to cancel the renormalization scale
+dependence introduced in the propagator pole by the run-
+ning Higgs mass computed in the effective potential ap-
+proach [15, 16].
+Finally, we point out that electroweak contributions at
+three-loop level is still missing, but the analytic results
+for all master integrals contributing to the three-loop
+Higgs self-energy diagrams in the mixed EW-QCD sec-
+tor at order α2αs and including terms proportional to the
+product of the bottom and top Yukawa couplings, ybyt,
+have been presented in [17]. Besides, additional identities
+satisfied by three-loop self-energy Master Integrals (MIs)
+with four and five propagators, which enable a straight-
+forward numerical evaluation for a generic configuration
+of the masses in the propagators, have been recently re-
+ported in [18].
+The paper is organized as follows. In Section II we show
+the technical details about the generation and regular-
+ization of the amplitudes for the three-loop Higgs self-
+energies involved in our calculation.
+In Section III a
+Feynman integral reduction procedure is presented and
+the election of a good basis of master integrals is dis-
+cussed. A numerical analysis, where the obtained three-
+loop corrections to the Higgs mass at O(y6
+t ) is evaluated
+as a function of the renormalization scale, is shown in
+Section IV. Finally, we give our conclusions and a fur-
+ther research outlook in Section V.
+1
+2
+3
+9
+6
+8
+4
+5
+7
+FIG. 1.
+Examples of diagrams contributing to the O(y6
+t )
+Higgs self-energy corrections in the non-QCD sector.
+The
+external dashed lines represent the Higgs boson field. The in-
+ternal dashed lines represent all possible contributing scalar
+fields, while the solid lines represent a top or a bottom quark.
+Only propagators with a top quark line are massive.
+II.
+REGULARIZED HIGGS SELF-ENERGIES
+In this work we have focused our attention on the con-
+tributions coming from the three-loop self-energy correc-
+tions to the Higgs boson mass including the external mo-
+mentum dependence. The Higgs self-energies have been
+computed at order y6
+t in the non-QCD sector of the SM.
+Thus, the non-light fermion limit is assumed and there-
+fore the Yukawa couplings and the masses of the other
+fermions are disregarded with respect to the top quark
+ones. The complete expression is written as
+Σ(3l)
+hh = y6
+t (∆0 + t∆1 + t2∆2 + t3∆3)
++ s y6
+t (∆s
+0 + t∆s
+1 + t2∆s
+2),
+(1)
+where t represents the squared top mass, t = M 2
+t , while
+s stands for the squared momentum in the external lines
+of the Higgs self-energies, s = p2.
+In order to obtain the expressions of ∆i and ∆s
+j it is
+necessary to generate the Higgs self-energy diagrams and
+their corresponding amplitudes. This has been done with
+the help of the Mathematica package FeynArts [19, 20].
+At the considered perturbative order, only the nine differ-
+ent self-energy topologies depicted in FIG. 1 contribute.
+Note that topologies with just cubic vertices are required,
+this is equivalent to impose an adjacency of three lines in
+the CreateTopologies function of FeynArts. Moreover,
+the computation was done in the so called Parameter
+Renormalized Tadpole (PRT) scheme [10, 21–23] where
+the renormalized vacuum expectation value of the Higgs
+field is the minimum of the Higgs effective potential and
+therefore the self-energies are made of 1PI diagrams that
+do not contain tadpole insertions. Although this scheme
+is known to be numerical stable as terms with negative
+powers of the Higgs self-coupling are not included, it
+has the unpleasant feature that self-energies are gauge-
+dependent quantities. In this work we have adopted the
+
+3
+Landau gauge, where the Goldstone bosons are massless,
+in order to minimize the number of energy scales appear-
+ing in the Feynman integrals.
+Once the content of the particles are included in the nine
+topologies, with the help of the InsertFields function of
+FeynArts, the number of generated self-energy diagrams,
+whose amplitudes are different from zero at y6
+t , increases
+to 125.
+Examples of such diagrams are also shown in
+FIG. 1. Note that the external dashed lines propagate
+only the Higgs field (h), while the internal lines in the
+no-light fermions limit of the non-QCD sector can prop-
+agate fermions (solid lines) like the top quark (t) and
+bottom quark (b) fields, as well as scalars like the Higgs
+and the Goldstone bosons (G0 and G±) fields. The cubic
+vertices involved in the computation are hht, G0G0t and
+G±tb. The contribution of the bottom mass to the latter
+vertex is disregarded when appears in the numerators of
+the integrands.
+The considered three-loop self-energy integrals are ul-
+traviolet divergent in four-dimensions since all of them
+contain two scalar and six fermionic propagators; there-
+fore, they are analytically continued to D = 4 − 2ε di-
+mensions using the dimensional regularization (DREG)
+scheme [24–27].
+In order to implement the regular-
+ization prescription, the FeynArts amplitudes are ex-
+ported to the language of FeynCalc [28, 29] which is a
+Mathematica code useful in general to perform the nec-
+essary algebraically manipulations involved in the cal-
+culation of multi-loop Feynman integrals. The gamma
+matrices are defined as a set of D matrices obeying
+{γµ, γν} = 2gµνI;
+TrI = 4.
+(2)
+Feynman diagrams involving the charged Goldstone
+bosons, G±, where traces with γ5 and an arbitrary num-
+ber of gamma matrices appear, require some care.
+In
+that case we use the practical non-cyclicity prescrip-
+tion [30, 31] where γ5 is an anticommuting object sat-
+isfying
+{γ5, γµ} = 0;
+γ2
+5 = 1,
+(3)
+together with the condition that the use of cyclicity in
+traces involving an odd number of γ5 matrices is forbid-
+den. Using the above anticommutation relation and the
+Clifford algebra in eq. (2) any product of Dirac matrices
+can be ordered in a canonical way. That is, the γ5 ma-
+trices are completely eliminated for an even number of
+them, while for an odd number only one γ5 survives and
+it is always moved to the right of the product. In partic-
+ular, due to the presence of four independent momentum
+scales, namely, the external momentum p and the loop-
+momenta q1, q2 and q3, diagrams can contain traces with
+a single γ5 and at most four γ matrices. Thus, the next
+relations are also required:
+Tr [γ5] = Tr [γµ1 . . . γµ2n−1γ5] = 0,
+(4)
+Tr [γµγνγργσγ5] =
+�
+−4iϵµνρσ
+µ, ν, ρ, σ ∈ {0, 1, 2, 3}
+0
+otherwise
+.
+(5)
+A further examination of all the Feynman diagrams for
+each topology in FIG. 1 shows that topologies 1, 4, 6 and
+9 do not contain traces with the matrix γ5. Topologies
+5 and 8 contain traces with one γ5 and at most three
+γ matrices which vanish according to eq. (4). For the
+topologies 2 and 7 the sum of the amplitudes produces
+a cancellation of the terms with any trace involving the
+matrix γ5. Finally, topology 3 contain contributions with
+a trace of a single γ5 and four γ matrices that have to be
+evaluated according to eq. (5).
+In addition, it is worth mentioning that for amplitudes
+with closed fermion-loops, which is the case of all the
+topologies in FIG. 1, the usual Breitenlohner-Maison
+scheme [32, 33] and the non-cyclicity scheme considered
+in our calculation produce identical results.
+III.
+GOOD MASTER INTEGRALS
+Once the amplitudes are regularized, each of them can
+be written as a superposition of a large set of about one
+thousand of integrals with the following structure:
+�
+N(qi · qj, qi · p, p2)
+Dν1
+1 Dν2
+2 Dν3
+3 Dν4
+4 Dν5
+5 Dν6
+6 Dν7
+7 Dν8
+8 Dν9
+9 Dν0
+0
+�
+3l
+,
+(6)
+⟨(. . . )⟩3l = (Q2)3ε
+�
+dDq1
+(2π)D
+�
+dDq2
+(2π)D
+�
+dDq3
+(2π)D ,
+where Q is the renormalization scale defined as in the MS
+scheme, Q2 = 4πe−γEµ2, in terms of the unit mass µ and
+of the Euler-Mascheroni constant γE. The denominators
+Dj are inverse scalar propagators:
+D1 =
+�
+q2
+1 − m2
+1
+�
+,
+D2 =
+�
+q2
+2 − m2
+2
+�
+,
+D3 =
+�
+q2
+3 − m2
+3
+�
+,
+D4 =
+�
+(q1 − q2)2 − m2
+4
+�
+,
+D5 =
+�
+(q1 − q3)2 − m2
+5
+�
+,
+D6 =
+�
+(q2 − q3)2 − m2
+6
+�
+,
+D7 =
+�
+(q1 + p)2 − m2
+7
+�
+,
+D8 =
+�
+(q2 + p)2 − m2
+8
+�
+,
+D9 =
+�
+(q3 + p)2 − m2
+9
+�
+,
+D0 =
+�
+(q1 − q2 + q3 + p)2 − m2
+0
+�
+,
+(7)
+while the numerator N is a function of scalar products
+involving the three loop momenta and the external mo-
+menta. At this point the coefficients of the integrals de-
+pend on yt, t and s, while the masses in the propagators
+D−1
+j
+can be mj = 0, Mt. The precise configuration of
+the masses defines the family to which the integrals be-
+long, while the set of exponents {νj} defines sectors from
+the families. For the planar diagrams, represented by the
+topologies 1 to 8, one must remove the denominator D0
+which is equivalent to set ν0 = 0, while the non-planar di-
+agrams contained in the topology 9 satisfy ν8 = 0. Note
+that, in order to express any scalar product in N as a
+combination of inverse propagators, we need a basis of
+nine propagators for each family. Thus, the numerator
+N is rewritten, as usual, in terms of the Dj’s leading to
+scalar integrals which can also contain irreducible numer-
+ators, that is, denominators with negative integer expo-
+nents. The resulting integral families for each topology
+are listed in Table I. An individual topology can contain
+
+4
+Topology
+Propagator
+1
+{134679}
+2
+{1278}, {12378}
+3
+{1379}, {123789}, {134679}
+4
+{24589}
+5
+{258}, {278}, {2578}, {24589}
+6
+{125678}
+7
+{17}, {147}, {157}, {1457}
+8
+{17}, {127}, {157}, {1257}
+9
+{123790}
+TABLE I. Integral families. An integral family is represented
+with a list {ijk...}. Each number in the list gives the position
+“j” of a massive propagator D−1
+j . The missing propagators
+are massless.
+multiple families and each family can contain at most six
+massive propagators. Besides, the exponents {νj} take
+values from −3 to 3.
+The obtained set of scalar integrals are not independent
+of each other, they can be related through additional re-
+currence relations coming from the integration by parts
+(IBP) and Lorentz Invariant (LI) identities.
+We have
+used the code Reduze [34, 35] to reduce any scalar inte-
+gral as a linear superposition of a basis of Master Inte-
+grals
+˜G{ν0,...,ν9} =
+� 9
+�
+j=0
+D−νj
+j
+�
+3l
+,
+(8)
+with coefficients that are rational functions of polyno-
+mials depending on the space-time dimension and all the
+kinematical invariants involved in the calculation. As ex-
+pected, in complicated situations like the IBP reduction
+of three-loop self-energy integrals with at least two en-
+ergy scales, the basis provided by Reduze, ˜G{i}, can be
+inefficient since denominators of some of the MIs coeffi-
+cients are quite cumbersome, containing big expressions
+that require a long time processing and operative mem-
+ory, but also containing kinematical singularities (inde-
+pendent upon D) described by the Landau conditions
+[36] and/or divergences in D−4 = 2ε (independent upon
+the kinematical invariants) which would imply the eval-
+uation of finite parts of the Laurent expansion in ε of
+the MIs [37–39]. In order to handle this situation we fol-
+low the prescription discussed in [40] based on the Sab-
+bah’s theorem [41] and therefore we have implemented in
+Mathematica, with the help of FIRE [42, 43], a transition
+from the “bad” basis of MIs, to an appropriate basis,
+G{j}, where denominators of the coefficients are “good”
+enough that are simple expressions free of kinematical
+and non-kinematical singularities. Thus, the election of
+the new master integrals has been done by imposing that
+polynomials in the denominators of the coefficients do not
+vanish in the limit where D − 4 goes to zero. The Sab-
+bah’s theorem guarantees the existence of such a good
+basis, but in practice this implies finding extra relations
+between the master integrals, such that
+˜G{i} =
+|σ|
+�
+j=1
+ni,j
+di,j
+G{j},
+(9)
+for a given sector σ of which |σ| represents the length
+of the related multi-index, and where the coefficients ni,j
+must contain products of polynomials that cancel the bad
+denominators of the coefficients of the masters ˜G{i} in
+the original IBP reduction, while di,j must be a good de-
+nominator. A simple example can be found in the family
+{134679} of the first topology (see FIG. 1 and Table I).
+A bad election of the basis in the reduction procedure
+can lead to coefficients with nul denominators for D = 4,
+of the form
+(−5 + D)(−4 + D)(−3 + D)(−10 + 3D)st2(−s + 2t)
+×(−s + 4t)(−s + 10t)(s2 − 16st + 24t2)
+(10)
+or an even worse coefficient can arise with denominator
+2(−4 + D)(s − 4t)2t(−38997504s18 + 159422976Ds18)
+×t(−288550464D2s18 + · · · + 244 terms), (11)
+manifesting moreover threshold singularities.
+The de-
+nominator of eq. (11) is generated by the sector
+with the MIs ˜G{−1,0,1,1,0,1,1,0,0},
+˜G{0,0,2,1,0,2,1,0,0} and
+˜G{0,0,1,1,0,1,1,0,0} .
+A better choice of the basis, with
+the master integrals G{1,−1,1,1,1,1,1,1,0}, G{1,0,1,1,1,1,1,1,1},
+G{0,0,1,1,1,1,1,1,1}, can avoid this problem and produce a
+simpler result of the total amplitude for the first topol-
+ogy:
+A
+{134679}
+1
+= y6
+t
+�
+t
+�
+4G{0,0,1,1,1,0,1,1,1} + 2G{0,0,1,1,1,1,0,1,1}
+− 4G{0,0,1,1,1,1,1,0,1} − 4G{1,−1,1,1,0,1,1,1,1}
++ 2G{1,−1,1,1,1,1,0,1,1} + 2G{1,−1,1,1,1,1,1,1,0}
++ 4G{1,0,0,0,1,1,1,1,1} − 4G{1,0,0,1,1,1,1,0,1}
++ 2G{1,0,0,1,1,1,1,1,0} − 4G{1,0,1,1,0,1,0,1,1}
++ 4G{1,0,1,1,0,1,1,0,1} − 4G{1,0,1,1,0,1,1,1,0}
++ 2G{1,0,1,1,1,1,−1,1,1} − 2G{1,0,1,1,1,1,0,0,1}
+−2G{1,0,1,1,1,1,1,0,0} + 2G{1,0,1,1,1,1,1,1,−1}
+�
++ t2 �
+8G{0,0,1,1,1,1,1,1,1} + 8G{1,0,0,1,1,1,1,1,1}
++ 8G{1,0,1,0,1,1,1,1,1} − 16G{1,0,1,1,0,1,1,1,1}
++ 8G{1,0,1,1,1,0,1,1,1} + 8G{1,0,1,1,1,1,0,1,1}
+−16G{1,0,1,1,1,1,1,0,1} + 8G{1,0,1,1,1,1,1,1,0}
+�
++ t3 32G{1,0,1,1,1,1,1,1,1}
+�
++ sy6
+t
+�
+t
+�
+−2G{1,0,1,0,1,1,1,1,1} + 4G{1,0,1,1,0,1,1,1,1}
+− 2G{1,0,1,1,1,0,1,1,1} − 2G{1,0,1,1,1,1,0,1,1}
++4G{1,0,1,1,1,1,1,0,1} − 2G{1,0,1,1,1,1,1,1,0}
+�
+−t2 8G{1,0,1,1,1,1,1,1,1}
+�
+(12)
+without pathological denominators.
+Note that master
+integrals contain 9 indices because D0 is omitted in the
+
+5
+planar topologies while D8 is removed in non-planar dia-
+grams. Analogous simple expressions have been derived
+for topologies 2, 4 and 6, the results for the amplitudes
+A3, A5, A7, A8 and A9 are instead somewhat lengthy.
+All the amplitudes can be consulted by the following
+link https://github.com/fisicateoricaUDP/HiggsSM
+together with the list of good master integrals, the useful
+IBP reductions and the main Mathematica routines im-
+plemented to carry out this computation. In particular,
+the planar diagrams can be reduced to a superposition
+of 212 MIs, while the non-planar diagrams can be ex-
+pressed in terms of 82 masters. Even if a good basis of
+MIs could be found with the help of the Sabbah’s the-
+orem in this computation, when the number of energy
+scales is increased the coefficients of the master integrals
+get even worse and make inefficient any IBP reduction
+procedure. This kind of problems also appears in beyond
+the SM theories, as is the case of the SUSY calculations
+of Mh, where the analogous contribution at order y6
+t is
+missing [44] and at least one additional scale (the squarks
+mass scale) has to be included. Analytical approaches
+where an IBP reduction can be avoided and the ampli-
+tudes can be directly evaluated for an arbitrary number
+of energy scales, as it is done for instance with the Loop-
+Tree Duality technique [45, 46], could be an interesting
+alternative.
+IV.
+NUMERICAL ANALYSIS
+In this section we discuss the numerical evaluation of
+the three-loop Higgs self-energy corrections at O(y6
+t ) ob-
+tained after summing the amplitudes Aj of the 21 fami-
+lies reported in Table I. The final amplitude of the gen-
+uine three-loop 1PI Higgs self energy
+Σ(3l)
+hh (s, Q, Mt, yt) =
+�
+j
+Aj,
+(13)
+requires the evaluation of 294 MIs which are functions
+of the top quark mass Mt and the squared external
+momentum s of the self-energies. We set the value of the
+external momentum at the experimental central value of
+the Higgs boson mass Mh, √s = 125.09 GeV [47]. In
+order to numerically generate the Laurent ε-expansion
+of each master integral, we have used the code FIESTA
+5.0 [48] which implements the sector decomposition
+approach.
+The expansion goes up to ε0 order, the
+evanescent terms of order εn with n > 0 are not needed
+since the coefficients of the good master integrals do
+not contains poles in D = 4. Besides, the evaluation of
+the amplitude has to include the evolution of the top
+Yukawa coupling yt and the mass parameter Mt as a
+function of the energy scale Q in the MS scheme.
+In this analysis we use the full three-loop MS renor-
+malization group equations (RGEs) of the SM param-
+eters [49–56] plus the O(α5
+s) QCD contributions to the
+strong coupling beta function [57–60] and the O(α5
+s)
+QCD contributions to the beta functions of the Yukawa
+5-Loops
+100
+200
+300
+400
+500
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+Q(GeV)
+yt
+Q0 = 0.1731 TeV
+αs = 0.107551
+yt = 0.934801
+g1 = 0.647660
+g2 = 0.358539
+v = 246.6 GeV
+λ = 0.126038
+FIG. 2. Renormalization group evolution of the top Yukawa
+coupling yt in the MS scheme including the full 3-loop RGEs
+for all the SM parameters and the QCD beta functions of yt
+and αs up to 5-loops. Here g1 and g2 stands for the EW gauge
+couplings, v is the Higgs vev and λ represents the quartic
+Higgs self-coupling.
+couplings [61–63]. This is in order to obtain the running
+of yt from 10 to 500 GeV as is shown in FIG. 2. To draw
+the evolution we chose the initial benchmark model
+point specified on the top of the plot, which yields at
+Q0 = 0.1731 TeV the central values of the SM masses
+(Mh = 125.1 GeV, Mt = 173.1 GeV, etc.) as given in
+the last edition of the Review of Particle Properties [64].
+The next plots also follows this boundary condition.
+On the other hand,
+the top quark pole mass is
+evolved in the MS/PRT scheme with the help of SMDR,
+as is shown in the FIG. 3, including the pure QCD
+1-loop [65], 2-loop [66], 3-loop [67] and 4-loop [68, 69]
+contributions plus the non-QCD 1-loop [70], mixed
+EW-QCD 2-loop [71] and full 2-loop EW [72] corrections
+to the quark top mass. The black curve contains all the
+1L QCD
+2L QCD
+3L QCD
+4L QCD
+4L+1L nQCD
+4L+2L Mixed
+4L+2L Full
+20
+50
+100
+200
+500
+171
+172
+173
+174
+175
+Q(GeV)
+Mt(GeV)
+FIG. 3. Evolution of the top quark mass Mt as a function of
+the renormalization scale Q in the MS scheme. The differ-
+ent perturbative contributions are shown. In particular, the
+black line contains the 4-loop QCD and the full 2-loop EW
+corrections.
+
+6
+contributions together, while the other lines represent
+the theoretical predictions of Mt at different perturba-
+tive orders. Note that the pure QCD predictions have
+a very large scale dependence of a few GeVs when Q
+is varied from 60 to 500 GeV and therefore the EW
+corrections cannot be neglected and must be included
+in the numerical analysis since our amplitudes are
+sensible to the precise value of Mt. When the full 2-loop
+EW contribution is added, the renormalization scale
+dependence decreases by about 97% in the range of Q
+considered.
+Finally,
+we study the numerical behaviour of the
+resulting new contributions to the Higgs self-energies
+containing all momentum dependence which are obtained
+from the difference
+∆Mh = Re
+�
+Σ(3l)
+hh (p2 = M 2
+h) − Σ(3l)
+hh (p2 = 0)
+�
+.
+(14)
+In FIG. 4 ∆Mh is shown as a function of the renormal-
+ization scale from Q = 60 GeV to Q = 500 GeV. In the
+plot is included the real contributions from the finite part
+(black curve) and the coefficients of the simple (yellow)
+1
+ε, double (green)
+1
+ε2 and triple (red)
+1
+ε3 poles separately.
+Note from FIG. 2 that the coupling yt goes out the per-
+turbative regime bellow Q = 60 GeV and therefore this
+region was excluded in the analysis. The coefficients of
+Finite
+Simple
+Double
+Triple
+100
+200
+300
+400
+500
+0
+10
+20
+30
+40
+50
+60
+70
+Q(GeV)
+ΔMh(MeV)
+FIG. 4. Renormalization group scale dependence coming from
+the external momentum contribution to the three-loop Higgs
+self-energy correction at order y6
+t in the SM. The evolution of
+the finite part and the coefficients of the simple, double and
+triple poles have been included.
+the poles have a mild dependence on the renormalization
+scale, the triple pole coefficient varies about 0.5 MeV for
+60 GeV ≤ Q ≤ 500 GeV, in this case the dependence on
+Q is not explicit, the variation is due to the RG evolution
+of yt and Mt. The double pole coefficient contains an ex-
+plicit logarithmic dependence on Q implying a variation
+of about 1.5 MeV. The simple pole coefficient contains a
+squared logarithmic dependence on Q which amounts to
+a variation of about 6.2 MeV. Finally, the finite part have
+a size of about 51 MeV for Q = 173.1 GeV and contains a
+significant renormalization scale dependence, it decreases
+by about 47% in the complete Q range considered. In
+particular, when Q is varied around the EW scale from
+100 GeV to 300 GeV the correction is reduced by about
+16 MeV which is of the same order of magnitude than
+the size of the anticipated experimental precision at HL-
+LHC (10 − 20 MeV [73]) and at the future colliders ILC
+(14 MeV [74]) and FCC-ee (11 MeV [75]). The inclusion
+of the new three-loop corrections ∆Mh into the complex
+pole mass, sh
+pole, for the SM Higgs boson and the fur-
+ther analysis of the numerical impact on the theoretical
+prediction of the Higgs boson pole mass are non-trivial
+tasks. They require the iterative evaluation of the MIs
+and amplitudes at s = Re(sh
+pole), instead of the naive
+evaluation at s = M 2
+h, and an additional prescription for
+the renormalization of the UV sub-divergences in order to
+get the correct values of the Mh-predictions at three-loop
+level. The numerical evaluation of the Higgs boson pole
+mass, including the pure three-loop corrections presented
+in this work, will be done in a future analysis.
+V.
+CONCLUSIONS AND PERSPECTIVES
+In this article we have presented a new contribution
+to the SM Higgs boson mass perturbative corrections
+coming from the pure three-loop Higgs self-energies at
+order y6
+t including the external momentum dependence.
+This implies a Feynman diagrammatic evaluation of eight
+planar and one non-planar topologies with only cubic
+vertices and a fermion loop in the internal lines.
+The
+Higgs self-energies do not contain the tadpole contribu-
+tions since the renormalized vev of the Higgs field is con-
+sidered as the minimum of the Higgs effective potential.
+As a consequence, the considered contributions have a
+good perturbative behaviour but acquire an additional
+gauge dependence, we have used the Landau gauge in
+order to reduce the number of energy scales in the Feyn-
+man amplitudes. Besides, we worked in the gaugeless and
+non-light fermions limits where the EW vector boson and
+all the light fermion masses are disregarded; thus, the fi-
+nal result is expressed in terms of the top quark mass Mt
+and the Higgs boson mass Mh. The DREG procedure
+was adopted in order to regularize the Feynman ampli-
+tudes associated to the Higgs self-energies, in particular,
+a non-ciclicity prescription was applied to deal with the
+regularization of the γ5 matrix. The resulting regular-
+ized amplitudes are expressed in terms of thousands of
+scalar integrals which are reduced to a superposition of
+a basis of master integrals through the IBP and LI iden-
+tities implemented in the code Reduze. This automated
+reduction leads to a set of master integrals which con-
+tains large coefficients with kinematic singularities and
+non-kinematic divergences at D = 4 space-time dimen-
+sions. The above mentioned singular behaviour as well as
+the length of the expressions of the coefficients get worse
+when the number of scales is increased.
+However, we
+have showed that those divergences are spurious and can
+be removed with a good redefinition of a suitable basis,
+
+7
+whose existence is guarantied by the Sabbah’s theorem.
+The expressions obtained for the amplitudes of the in-
+volved topologies are thus linear combinations of a set of
+212 planar and 82 non-planar good MIs with coefficients
+that do not contain poles at D → 4, it has the advantage
+that the evanescent terms of the Laurent expansion of
+the masters are not required. A first numerical analysis
+allows to measure the size of the new momentum depen-
+dent Higgs self-energy contributions showing a value of
+∼ 51 MeV at the benchmark model point which produces
+the experimental values of the SM masses, but it also dis-
+plays a significant renormalization scale dependence of a
+few tens of MeV which are of the same magnitude order
+than the expected precision at the coming colliders ex-
+periments.
+Several research perspectives are left open for future
+works. The inclusion of the new momentum dependent
+corrections into the complex mass pole of the Higgs prop-
+agator and the study of the numerical impact on the theo-
+retical prediction together with the perturbative stability
+of MS renormalization of the Higgs mass will be faced in
+a forthcoming publication. Besides, the developed rou-
+tines for this computation will be extended to include
+the quantum corrections to the SM gauge boson masses
+MZ and MW at the same perturbative order considered
+here. An extension of the momentum dependent Higgs
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+t to include supersymmetric con-
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+arXiv:2301.13634v1  [cs.CE]  16 Dec 2022
+Reduced order modelling using parameterized
+non-uniform boundary conditions in room acoustic
+simulations
+Hermes Sampedro Llopis,1, 2, a Cheol-Ho Jeong,1 and Allan P. Engsig-Karup3
+1Acoustic Technology Group, Department of Electrical and Photonics Engineering, Technical Uni-
+versity of Denmark, Kongens Lyngby, Denmark
+2Rambøll Denmark, Copenhagen, Denmark
+3Scientific Computing Section, Department of Applied Mathematics and Computer Science, Techni-
+cal University of Denmark, Kongens Lyngby, Denmark
+(Dated: 1 February 2023)
+Quick simulations for iterative evaluations of multi-design variables and boundary conditions
+are essential to find the optimal acoustic conditions in building design. We propose to use the
+reduced basis method (RBM) for realistic room acoustic scenarios where the surfaces have
+inhomogeneous acoustic properties, which enables quick evaluations of changing absorption
+materials for different surfaces in room acoustic simulations. The RBM has shown its benefit
+to speed up room acoustic simulations by three orders of magnitude for uniform boundary
+conditions. This study investigates the RBM with two main focuses, 1) various source posi-
+tions in diverse geometries, e.g., square, rectangular, L-shaped, and disproportionate room.
+2) Inhomogeneous surface absorption in 2D and 3D by parameterizing numerous acoustic pa-
+rameters of surfaces, e.g., the thickness of a porous material, cavity depth, switching between
+a frequency independent (e.g., hard surface) and frequency dependent boundary condition.
+Results of numerical experiments show speedups of more than two orders of magnitude com-
+pared to a high fidelity numerical solver in a 3D case where reverberation time varies within
+one just noticeable difference in all the frequency octave bands.
+[https://doi.org(DOI number)]
+[XYZ]
+Pages: 1–11
+I. INTRODUCTION
+Room acoustic simulations are typically used during
+the design stage of a building to find the optimal amount,
+choice and position of materials according to the use of
+the room. Poor acoustic conditions can produce a neg-
+ative effect, e.g., a decrease in the work productivity1,2,
+a decrease in the learning quality3,4, and an increase in
+the stress level5,6. Performing room acoustic simulations
+by solving the acoustic wave equation using numerical
+methods, such as Finite Difference Time-Domain meth-
+ods (FDTD)7, Finite Element Methods (FEM)8 or Spec-
+tral Element Methods (SEM)9 is accurate as all impor-
+tant wave phenomena can be accounted for. However,
+it is computationally expensive compared to geometric
+acoustics methods10. As a result, active research seeks
+to find ways to speed up the numerical methods to be
+used in building design. Common strategies include us-
+ing model order reduction (MOR)11, parallel computing
+on multi-core processing units12, and more recently also
+machine learning techniques13.
+A review of the state-of-the-art in MOR across dis-
+ciplines can be found in the literature14. The reduced
+ahsllo@elektro.dtu.dk
+basis method (RBM)11 is a method belonging to the
+class of MOR techniques. It exploits the parametric de-
+pendence in the solution of a partial differential equa-
+tion (PDE) by combining different solutions given by
+the variation of a set of parameter values.
+Problems
+of large systems of PDEs are effectively reduced to a
+low dimensional subspace to achieve computational ac-
+celeration and transformed back to the original problem
+size15–22, however, the speedup is dependent on the prob-
+lem of interest. MOR have been recently presented in
+some acoustic applications23–25 as well as in many dif-
+ferent fields, e.g., electromagnetics26,27, computational
+fluid dynamics28,29, heat transfer30, vibroacoustics31–33
+and many others34–38, demonstrating, in general, an ef-
+ficient reduction in the computational burden.
+Using
+RBM in room acoustic simulations with parameterized
+boundary conditions during the design stage of an in-
+door space may allow exploring many acoustic conditions
+by solving the reduced system and exploiting the benefit
+of a reduced computational cost under the variation of
+several parameters, e.g., the thickness of a porous ma-
+terial, air gap distance, etc. Today, the application of
+MOR to room acoustic simulations with boundary pa-
+rameterization is scarce despite the significant potential
+for acceleration. A recent study39 presents a model order
+reduction strategy using a Krylov subspace algorithm in
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+1
+
+the time domain with a FEM solver where speedups of
+11–36 were demonstrated. The study is performed for
+a simple domain where all the surfaces are assumed to
+be rigid boundaries, and only the floor is modelled with
+a surface impedance. Moreover, another study demon-
+strates the potential of RBM in room acoustic applica-
+tions, however, only for simplified homogeneous bound-
+ary conditions40. Two to three orders of speedup factors
+were achieved in the online stage.
+The study of RBM with realistic inhomogeneous
+boundary conditions, where the absorption materials
+are distributed inhomogeneously among the surfaces, is
+scarce in the literature. For example, classrooms typi-
+cally have highly absorbing ceilings and scattering ob-
+jects near the rear surface and window on one side wall.
+In this study, we investigate how to effectively apply
+RBM in a realistic setting with inhomogeneous boundary
+conditions, how the RBM performance varies with the
+room geometry and source/receiver location, and what
+acceleration is expected in such conditions.
+The novelty of this investigation is to construct and
+analyze the performance of a RBM strategy for realis-
+tic scenarios with two focuses, one being the RBM per-
+formance in various room shapes and source locations
+and two being the performance when including numerous
+parameters of acoustic materials distributed inhomoge-
+neously. First, this study presents a conceptual proof-of-
+concept for a complex 2D case, as it simplifies the com-
+putational burden. Second, two 3D rooms are analyzed.
+This study is essential for future work, e.g., scaling the
+method to extend the RBM for large building projects.
+In Sec.
+II the governing equations, the boundary
+conditions, the reduced basis method and the error mea-
+sures are described together with the different domains
+and simulation parameters. Section III handles the nu-
+merical experiments and results, which are analyzed and
+discussed in Sec. IV.
+II. METHODS
+This section presents an overview of the methods and
+simulation conditions used in this study. A more detailed
+description of the full order model (FOM) solver used as
+a reference model and the ROM is deeply described in
+previous work40.
+A. Governing equations and boundary conditions
+We consider the acoustic wave equation in the
+Laplace domain
+s2p − c2∆p = 0,
+(1)
+where p(x, t) is the sound pressure, x ∈ Ω the po-
+sition in the domain Ω ⊂ Rd with d = {2, 3}, t is the
+time in the interval (0, T ] s and c is the speed of sound
+(c = 343 m/s). Equation (1) is discretized using the SEM
+formulation, which is well known and an overview can be
+found9,41,42. The final formulation written in the Laplace
+domain is given by
+�
+s2M + c2S + sc2 ρ
+Zs
+MΓ
+�
+p = 0,
+(2)
+where M refers to the mass matrix, S is the stiffness
+matrix, ρ is the density of the medium (ρ
+=
+1.2
+kg/m3) and Zs is the surface impedance.
+Note that
+the impedance boundaries are considered denoting the
+boundary domain as Γ.
+The frequency dependent boundary conditions are
+implemented via the method of auxiliary differential
+equations (ADE)9,43,44.
+The surface impedance of a
+porous absorber is modelled using Miki’s model45 in con-
+junction with a transfer matrix method46, and mapped
+to a six pole rational function by using a vector fitting
+algorithm47 so that the surface admittance Ys = 1/Zs
+can be written as a rational function and expressed us-
+ing partial fraction decomposition44. Then, the system
+(2) can be stated in the form of a linear system of equa-
+tions and solved using a sparse direct solver as presented
+in9,40
+Kp = 0,
+K ∈ RN×N,
+(3)
+where, K refers to the operators shown in (2) and N
+corresponds to the degree of freedom (DOF). Note that
+if the system is split into real and imaginary parts for
+implementation purposes, the size of the operator is
+K ∈ R2N×2N 40,48. The system (2) is initialized using a
+Gaussian pulse with a spatial distribution σg that deter-
+mines the frequencies to span, by adding the right hand
+side term sMp0, where p0 is the initial sound pressure
+state in the time domain.
+The solution in the Laplace domain is finally trans-
+formed to the time domain by means of the Weeks
+method40,49.
+B. The reduced order model
+The purpose of using RBM is to substantially reduce
+the size of the problem while ensuring a certain level of
+accuracy.
+Specifically, the DOF are reduced, and the
+techniques succeed when RDOF ≪ DOF, RDOF being
+the corresponding degrees of freedom in the numerical
+scheme after applying RBM. The RBM consists of two
+stages. First, one or more variables present in the par-
+tial differential equation or its discretized form (2) are
+chosen as parameters, e.g., Zs, spanning a discrete range
+of values. Then, in the first stage, referred to as the of-
+fline stage, the parameter space is explored to generate a
+problem-dependent basis by collecting FOM solutions for
+different parameter values within the range of interest. A
+Galerkin projection takes place to reduce the dimension-
+ality of the problem by utilizing the generated basis. In
+the second stage, referred to as the online stage, the re-
+duced problem is solved for a new parameter value that
+was not explored in the offline stage at a much lower
+computational cost. The offline stage is typically com-
+putationally costly as it requires multiple FOM solutions
+2
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+
+that capture relevant information on the parameters vari-
+ations and allow generating of representative basis func-
+tions of that variation. The reduction of the size of the
+computational problem comes with the truncation of the
+basis. The solver is stated in the Laplace domain to en-
+sure the stability of the ROM solution48. The basis gen-
+eration can be seen as a data-driven technique based on
+proper orthogonal decomposition (POD). It relies on a
+proper symplectic decomposition (PSD) with a symplec-
+tic Galerkin projection.
+Specifically, the cotangent-lift
+method introduced in50 is applied, which preserves the
+structure of the operators when the problem is split and
+solved into real and imaginary parts. The ROM solution
+is expressed as an expansion of the basis functions φi and
+coefficients ai, which is represented as
+prom = Φa,
+(4)
+where Φij ≡ φi(xj). Inserting (4) into (3) yields to a
+similar problem, where now the system is solved for the
+coefficients a
+Kroma = 0,
+(5)
+where Krom = ΦT
+clKΦcl and Φcl defines the symplectic
+basis constructed as
+Φcl =
+�
+Φ 0
+0 Φ
+�
+.
+(6)
+Moreover, the reduced operator can be written also as
+Krom = s2ΦT MΦ + c2ΦT SΦ + sc2 ρ
+Zs
+ΦT MΓΦ,
+(7)
+where a new parameter value can be chosen during
+the online stage (for example, Zs).
+The generation
+of a basis comes by first collecting all the FOM solu-
+tions obtained during the offline stage in the snapshot
+matrix40 SN ∈ RN×2Ns, where N is already described
+above, and Ns is the number of evaluated complex
+frequencies determined by the Weeks method48.
+SN
+can be decomposed using the proper orthogonal decom-
+position (POD) technique based on a singular value
+decomposition (SVD) to get the corresponding basis
+functions, defined as Φ = [U1, ..., UNrb] ∈ CN×Nrb. The
+reduced basis is chosen through truncation of this basis
+relying on the rate of decay of the singular values.
+The singular value decay shows the energy distribu-
+tion among the basis, providing information about the
+reduction of the problem. It is defined as
+E/E0 = diag(σ1, .., σN)
+�N
+i=1 σi
+,
+(8)
+where Σ = diag(σ, .., σN ) is obtained by SVD where S =
+UΣV T . The number of basis can be chosen so that the
+projection error is smaller than a given tolerance ǫP OD
+I(Nrb) =
+�Nrb
+i=1 σ2
+i
+�N
+i=1 σ2
+i
+≥ 1 − ǫP OD,
+(9)
+where Nrb denotes the number of basis functions, I(Nrb)
+represents the percentage of the energy of the collection
+of FOM solutions captured by the first.
+For a given
+ǫP OD, the faster the energy decays, the smaller number
+of basis needed, and thus, a better reduction of the
+problem is expected. The reduction comes with a trun-
+cation of the basis, which determine the size of (7) as Nrb.
+C. Error measures
+In this study, two type of errors are considered to
+compare the ROM against the FOM. First, the relative
+error using the root mean square (rms) pressure is intro-
+duced
+ǫrel = prmsROM − prmsF OM
+prmsF OM
+× 100
+(%).
+(10)
+Second, the error in the frequency domain expressed in
+dBs is considered
+∆L(f) = 20 log10
+���pF OM(f)
+pROM(f)
+���,
+(11)
+where, pF OM and pROM are the sound pressure of the
+FOM and ROM respectively along the frequency spec-
+trum.
+The performance of the ROM is measured in terms
+of speedups (sp) defined as
+sp = CPUF OM
+CPUROM
+,
+(12)
+where CPUF OM and CPUROM corresponds to the com-
+putational time of the FOM and ROM respectively.
+D. Test rooms and simulation conditions
+First, Section III A, deals with ROMs with several
+2D geometries with different source locations illustrated
+in Figure 1.
+Four different geometries and two source
+positions are considered, one at the corner and one at
+the centre.
+First a 4 × 4 m2 square domain with the
+source placed at (sx1, sy1)SQ = (0.2, 0.2) m (SQ1), and
+(sx2, sy2)SQ = (2, 2) m (SQ2) is introduced (Figure
+1a).
+Second, a 4 × 2.5 m2 rectangular domain where
+(sx1, sy1)RC = (0.2, 0.2) m (RC1), and (sx2, sy2)SQ =
+(2, 1.25) m (RC2) is considered (Figure 1b). Third, an
+L-shaped room where the long side is 4 m and the short
+side is 2 m is considered with only one source position
+at the corner (sx, sy)LS = (0.2, 0.2) m (LS1) (Figure 1c).
+Finally a corridor shape of size 10 × 1 m2 is presented
+where (sx, sy)CO = (0.2, 0.2) m (CO1) (Figure 1d). The
+maximum element size is selected considering triangular
+high-order elements (P = 4) and using 4 points per wave-
+length (PPW) leading into an upper frequency fu = 2.8
+kHz, which is approximately the upper cutoff frequency
+of the 2 kHz octave band. The model is excited with a
+Gaussian pulse as initial condition with σg = 0.1 m251.
+The ROMs for each room type and source position are
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+3
+
+FIG. 1. The geometries and source positions.
+built by sampling the surface impedance of all the sides
+at the following values Zs = [500, 5250, 10000] Nsm−3.
+Second, section III B deals with several 2D ROMs for
+an inhomogeneous distribution of the acoustic material,
+which in this study is named inhomogeneous boundary
+conditions. A 2D rectangular room (4 m ×2.5 m) with
+inhomogeneous boundary conditions is considered with
+the same simulation parameters as before. The source
+is placed at (sx, sy) = (3, 1.2) m and the receiver is at
+(rx, ry) = (1, 1.2) m. Moreover, an additional ROM is
+constructed for this section with an upper frequency of
+fu = 4 kHz.
+Third, for section III C, two 3D models are consid-
+ered.
+A 1 m cube (CB) enclosure is presented, where
+three of the six surfaces are parameterized. Simulations
+are carried out using a polynomial order of P = 4 with
+N = 35937 elements.
+Assuming PPW = 4, the up-
+per frequency is given by fu = 2.8 kHz.
+The model
+is excited with a Gaussian pulse as an initial condition
+with σg = 0.1 m2 placed at (sx, sy, sz)CB = (0.7, 0.5, 0.5)
+m.
+The receiver position is placed at (rx, ry, rz) =
+(0.25, 0.25, 0.50) m. A second 3D room is considered to
+follow a good ratio (GR) of 1.9:1.4:1, which assures an
+even distribution of the room modes52. The room size
+is (Lx, Ly, Lz) = (1.615, 1.190, 0.850) m, sound source is
+placed at (sx, sy, sz)GR = (1.200, 0.600, 0.425) m and the
+receiver (rx, ry, rz) = (0.500, 0.200, 0.425) m. Again, a
+polynomial order of P = 4 with N = 35937 is considered.
+Assuming a spatial resolution corresponding to about 4
+PPW for the highest frequencies (fu = 1.7 kHz).
+E. Inhomogeneous boundary parameterization
+First, the 2D domain is considered in section III B.
+The ceiling (CE) is modelled with a porous absorber.
+The key parameters affecting the absorption character-
+istic of a porous absorber are the flow resistivity, thick-
+ness, and air cavity depth53. To parameterize the porous
+ceiling, FOMs with σmat = [10, 30, 50] kNsm−4, dmat =
+[0.02, 0.12, 0.22] m, and d0 = [0.02, 0.12, 0.22] m are sim-
+ulated in the offline stage. The floor (FL) is designed
+with two different options: as a hard surface modelled
+as a frequency independent boundary and covered with
+a carpet modelled as a porous layer.
+Frequency inde-
+102
+103
+Frequency [Hz]
+0
+0.2
+0.4
+0.6
+0.8
+1
+norm
+CEA
+CEB
+CEC
+CED
+(a)
+102
+103
+Frequency [Hz]
+0
+0.2
+0.4
+0.6
+0.8
+1
+norm
+FLA
+FLB
+FLC
+FLD
+FLE
+(b)
+102
+103
+Frequency [Hz]
+0
+0.2
+0.4
+0.6
+0.8
+1
+norm
+WA
+WB
+WC
+(c)
+FIG. 2. Absorption coefficients. a) Ceiling (CE), b) Floor
+(FL), c) Walls (W).
+pendent boundary conditions are ranges Zs = [10, 50, 90]
+kNsm−3, while frequency dependent boundary conditions
+to model the carpet are computed with a fixed thickness
+of 0.02 m but varying σmat = [10, 30, 50] kNsm−4. Fi-
+nally, the left wall (WL) and right (WR) walls are mod-
+elled as porous panels where dmat = 0.03 m, d0 = 0
+m and σmat = [5, 12, 19] kNsm−4. Figure 2 shows the
+absorption coefficients for the most and least absorptive
+cases of each surface, whose corresponding parameter val-
+ues are given in Table I.
+A way to construct the ROM is by performing
+FOM simulations for each combination of the parameter
+values. To cover the inhomogeneous boundary variation,
+a total of 37 (2187) FOM simulations are possible.
+This is way too many for practical runtime constraints.
+Instead, we have chosen only 3 × 7 (= 21) FOM sim-
+ulations, which is shown in the Appendix (Table A1)
+indicating for each FOM simulation which parameter
+values are chosen and which are fixed. The table rows
+correspond to the 21 simulations, and the columns
+correspond to the different parameter values shown in
+4
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+
+4m
+4
+3
+N
+3
+4m
+4m
+5
+13
+10mTABLE I. Parameter values of the presented absorption co-
+efficients of Figure 2.
+σmat [kNsm−4] dmat [m] d0 [m] Zs [kNsm−3]
+CEA
+10
+0.02
+0.02
+-
+CEB
+30
+0.12
+0.12
+-
+CEC
+10
+0.12
+0.22
+-
+CED
+30
+0.02
+0.22
+-
+FLA
+10
+0.02
+0
+-
+FLB
+30
+0.02
+0
+-
+FLC
+50
+0.02
+0
+-
+FLD
+-
+-
+-
+10
+FLE
+-
+-
+-
+90
+WA
+5
+0.03
+0
+-
+WB
+12
+0.03
+0
+-
+WC
+19
+0.03
+0
+-
+TABLE II. Parameter values chosen to construct the ROM in
+2D. The parameters under variation that are included in the
+ROM are marked with *.
+CE
+FL
+WL
+WR
+σmat [kNsm−4]
+[10, 30, 50]*
+[10, 30, 50]* [5, 12, 19]* [5, 12, 19]*
+dmat [m]
+[0.02, 0.12, 0.22]*
+0.02
+0.03
+0.03
+d0 [m]
+[0.02, 0.12, 0.22]*
+0
+0
+0
+Zs [kNsm−3]
+-
+[10, 50, 90]*
+-
+-
+Table II. The chosen parameters are marked with an
+X. To the author’s knowledge, this study is the first to
+report MOR performances with such complicated room
+acoustic setups and is, therefore, useful to establish the
+feasibility of the approach.
+Second, section III C considers the 3D domain for
+both cube and good ratio shapes. In both cases, the ceil-
+ing (CE) is modelled as a porous acoustic material of a
+fixed thickness dmat = 0.05 m, where σmat = [10, 30, 50]
+kNsm−4 and d0 = [0.02, 0.12, 0.22] m are parameterized.
+The floor (FL) is modelled as a frequency independent
+boundary with Zs = 50 kNsm−3. The east wall (WE)
+is modelled as a frequency independent brick surface
+with Zs = 50 kNsm−3.
+The south wall (WS) is not
+parameterized, and it is covered with a porous material
+where σmat = 7 kNsm−4, dmat = 0.02 m and d0 = 0 m.
+The west wall (WW) is modelled with a porous material
+of dmat = 0.05 m and d0 = 0 m whose flow resistivity is
+parameterized with values of σmat = [5, 12, 19] kNsm−4.
+The north wall (WN) is modelled as a hard surface with
+Zs = 50 kNsm−3.
+In addition, a 0.5 × 0.5 m2 square
+acoustic panel made of porous material is placed at the
+centre of the wall. The panel has a fixed thickness and
+flow resistivity of dmat = 0.1 m and σmat = 30 kNsm−4
+respectively.
+The air gap between the panel and the
+wall is parameterized d0 = [0.2, 0.12, 0.22] m. Table III
+summarizes the parameter values of each surface.
+TABLE III. Parameter values chosen to construct the ROM
+in 3D. The parameters under variation that are included in
+the ROM are marked with *.
+CE
+FL WE WS
+WW
+WN
+σmat [kNsm−4]
+[10, 30, 50]*
+-
+-
+70
+[5, 12, 19]*
+30
+dmat [m]
+0.05
+-
+-
+0.02
+0.05
+0.1
+d0 [m]
+[0.02, 0.12, 0.22]*
+-
+-
+0
+0
+[0.02, 0.12, 0.22]*
+Zs [kNsm−3]
+-
+50
+50
+-
+-
+-
+The ROM is constructed for each 3D room con-
+structed with four different parameters (2 for CE, 1 for
+WW and 1 for WN) with three different values each.
+Thus, a total number of 34 = 81 FOM simulations are
+possible.
+Instead, 3 × 4 = 12 FOM simulations were
+carried out to construct the ROM in the same way
+described for the 2D case. The chosen parameter values
+for each FOM simulation are shown in the Appendix
+(Table A2).
+III. RESULTS
+This section presents results mostly with the sin-
+gular energy decay, E/Eo, relative error shown in (10),
+speedups (12), sound pressure level (SPL) spectrum, and
+the reverberation time (RT).
+A. 2D - Influence of the source position and geometry
+For the geometries and source positions tested, the
+singular value decay is shown in Figure 3. A faster decay
+means that a larger portion of the energy is concentrated
+in the first singular values, effectively indicating that a
+smaller number of basis Nrb is needed for a given error
+tolerance. Slower decays would need a larger number of
+Nrb to provide the same error. Note that the smaller Nrb
+is, the higher speedups are achieved. In Figure 3, one
+can see that the more symmetric the problem is, both
+in terms of geometry and source location, the faster the
+decay is, as measured in the singular values translating
+into more efficiency (speedup). For example, SQ shows
+a faster decay than RC and LS.
+However, the differ-
+ence is not significant until the energy reaches the value
+of 10−10, and it can be concluded that the room geome-
+try does not significantly change the energy distribution
+among the basis. On the other hand, the centred sound
+source locations lead to a faster decay of the singular val-
+ues compared to the corner source. SQ2 shows a faster
+decay than SQ1, and the same for RC2 in comparison to
+RC1. This is because placing a source at the centre of
+the room will fail to excite some room modes, of which
+the nodal lines/points coincide with the source location.
+This leads to a smaller number of basis needed to de-
+scribe the physical dynamics of the wave propagation in
+the room accurately.
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+5
+
+0
+500
+1000
+1500
+2000
+2500
+Nrb
+10-20
+10-15
+10-10
+10-5
+100
+Singular values, | i|
+SQ1
+SQ2
+RC1
+RC2
+LS1
+CO1
+FIG. 3. Singular value decay for the first 2500 modes of the
+basis and energy distribution (E/E0) among the basis with
+frequency independent boundaries.
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Nrb
+10-20
+10-15
+10-10
+10-5
+100
+Singular values, | i|
+CE
+CE+FL
+CE+FL+WL
+CE+FL+WL+WR
+FIG. 4. Singular value decay for different number of param-
+eters for case 1 and case 2.
+B. 2D - Parameterization of different absorption properties
+In Figure 4, the singular value decay when a different
+number of parameters are included in the model, i.e.,
+only the ceiling parameters are included in the ROM
+(CE); the ceiling and the floor parameters (CE+FL); the
+ceiling, the floor and the left wall (CE+FL+WL) and
+all the sides (CE+FL+WL+WR). The more parameters
+added, the slower the decay curve, so adding more
+parameters to the ROM has a clear effect on the singular
+value decay and, thus, the choice of the basis for ROM.
+However, the decay remains similar in the first basis
+functions, which are the ones used to construct the ROM.
+In the online stage, two ROM cases are simulated and
+compared with their corresponding FOM. Case 1 deals
+with frequency dependent boundaries, while case 2 allows
+to change the floor to a rigid surface, modelled as a fre-
+quency independent boundary. The boundary parameter
+TABLE IV. Parameter values for the online stage of the 2D
+ROM. Values marked with * denotes the parameters which
+are parameterized.
+σmat [kNsm−4] dmat [m] d0 [m] Zs [kNsm−3]
+CE1
+2*
+0.1*
+0.1*
+-
+FL1
+12*
+0.02
+0
+-
+WL1
+10*
+0.03*
+0
+-
+WR1
+15*
+0.03
+0
+-
+CE2
+45*
+0.05*
+0.2*
+-
+FL2
+-
+-
+-
+7*
+WL2
+10*
+0.2*
+0
+-
+WR2
+6*
+-
+0
+-
+values are shown in Table IV, and the IR and SPL are
+shown in Figure 5. Several different online ROM simu-
+lations are compared for Nrb = 69, 158, 274, 410, 571, 752
+for the ROM with an upper frequency of 2 kHz corre-
+sponding to values of ǫP OD = 10−2, 10−3, ..., 10−7; and
+Nrb = 132, 310, 531, 810 for the ROM with an upper
+frequency of 4 kHz corresponding to values ǫP OD =
+10−2, ..., 10−5. Figure 5 shows the results up to 4 kHz
+with Nrb = 531. Figure 5(a) shows the impulse response,
+and Figure 5(b) presents the sound pressure level (SPL)
+from 20 Hz to 4 kHz showing that ∆L is nearly zero be-
+low 600 Hz and increasing with the frequency. Note that
+around 4 kHz, a roll-off is presented due to the source ex-
+citation. This case enables a computational speedup by
+a factor of 37 for an error of ǫrel = 0.03%. For the ROM
+constructed with Nrb = 274 and an upper frequency of 2
+kHz, the error is ǫrel = 0.2% for case 1 with a speedup
+of 70. Moreover, case 2 presents an error of ǫrel = 0.3%
+and a speedup of 50. Note that the accuracy depends on
+Nrb. The speedup against ǫrel given in (10) is shown in
+Figure 6. These results show speedups around two orders
+of magnitude for 2D.
+C. 3D - Parameterization of different absorption properties
+This section presents a similar analysis with 3D test
+cases. By varying Nrb, simulations are compared. The
+chosen parameters for the online simulation are shown
+in Table V. For the CB the following number of ba-
+sis are considered Nrb = 30, 81, 175, 275, 543, 837 corre-
+sponding to values of ǫP OD = 10−2, ..., 10−9 to be com-
+pared against the corresponding FOM solution for verifi-
+cation purposes. Moreover, for the GR room, the ROM
+was constructed with Nrb = 62, 303, 478, 649, 1500 corre-
+sponding to values of ǫP OD = 10−2, ..., 10−9. Table V
+shows the chosen parameter values for all the surfaces
+for both the cubic and good ratio domains. Figure 7(a)
+and Figure 7(b) present the impulse response and fre-
+quency response, respectively, for the CB. Moreover, Fig-
+ure 7(c) and Figure 7(d) show the impulse response and
+frequency response, respectively, for the GR room. In
+both cases, the ∆L is included, which confirms a good
+agreement between FOM and ROM for the given Nrb.
+The CB is constructed with Nrb = 273 where the error is
+6
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+
+0
+0.05
+0.1
+0.15
+0.2
+Time [s]
+-0.05
+0
+0.05
+0.1
+Pressure [Pa]
+ROM
+FOM
+(a)
+102
+103
+Frequency [Hz]
+-20
+-10
+0
+10
+20
+30
+SPL [dB]
+FOM
+ROM
+L
+(b)
+FIG. 5. (a) Impulse response and (b) spectrum of 2D FOM
+and ROM of case 1 for Nrb = 531 and fu = 4 kHz (using the
+parameter in Table 4).
+10-2
+10-1
+100
+101
+rel (%)
+101
+102
+103
+104
+speedup
+3DCB
+2D
+FIG. 6. Speedup against the relative rms error for the 2D
+domain (case 1) with fu = 2 kHz, and 3D CB.
+ǫrel = 0.04% with a speedup of 143. On the other hand,
+the GR room is constructed with Nrb = 478, where the
+error is ǫrel = 0.66% with a speedup of 90. Note ∆L
+in Eq. (11) and ǫrel are also presented for the different
+number of basis Nrb in Figure 8 for the CB (Figure 8(a))
+and the good ratio room (Figure 8(b)). Again, the ∆L
+is nearly zero at lower frequencies and increases with fre-
+quency, showing more differences at the anti-resonances.
+The speedup against the error for CB case 1 is presented
+in Figure 6, showing values around three orders of mag-
+nitude.
+In order to understand the behaviour of different
+room ratios, the number of basis functions Nrb for a given
+tolerance ǫP OD described in (9) is compared.
+A new
+ROM of the cubic room is computed with fu = 1.7 kHz.
+Figure 9 shows the comparison for GR1.7kHz, CB1.7kHz
+and CB2.8kHz. Results show that for a fixed upper fre-
+quency, a less symmetric geometry GR1.7kHz needs more
+basis functions for a given ǫP OD compared to a sym-
+metric geometry CB1.7kHz. This would not necessarily
+lead to lower speedups considering that GR has a larger
+number of DOF. Moreover, comparing the curves cor-
+responding to GR1.7kHz and CB2.8kHz for a fixed num-
+ber of DOF, GR1.7kHz results in a larger number of Nrb
+for a given ǫP OD. Thus, it will lead to lower speedups
+as the numerator of equation (12) remains the same for
+GR1.7kHz and CB2.8kHz. At the same time, the denomi-
+nator becomes larger for GR1.7kHz at a given ǫP OD com-
+pared to CB2.8kHz.
+The reverberation time is one of the most widely
+used acoustic parameters defined in ISO 3382-154 as the
+time needed for the energy to decrease by 60 dB. A way
+to evaluate the error of the reverberation time between
+the FOM and ROM is by means of the JND, which is
+the minimum change in the RT that can be perceptually
+perceived. The JND of the RT is 5% as defined in the
+standard54. Note that if the difference is larger than 1
+JND, the IR can be potentially differently heard.
+T20
+is calculated from IRs via FOM and ROM to quantify
+how ROM degrades the accuracy of RT. Figure 10(a)
+shows the RT for different frequency octave bands. The
+CB ROM is performed with Nrb = 185 while the GR
+ROM with Nrb = 649. The RT difference is below one
+JND in all the frequency octave bands in both cases,
+which indicates that the present ROM would not be
+perceptually different compared to the FOM. Moreover,
+Figure 10(b) and Figure 10(c) show the numbers of JND
+for T20 for various Nrb.
+Decreasing Nrb increases the
+RT difference in the higher frequency bands. Note that
+in this case the ROM is 365 times faster than the FOM
+and the CB needs fewer basis functions than GR, which
+supports the finding that symmetric conditions are more
+favourable for higher reductions.
+IV. DISCUSSION
+The performance behaviour of the ROM in terms
+of speedups is case-dependent and can be challenging to
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+7
+
+ROMFOM0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Time [s]
+-0.05
+0
+0.05
+IR [Pa]
+FOM
+ROM
+(a)
+102
+103
+Frequency [Hz]
+-40
+-20
+0
+20
+40
+SPL [dB]
+FOM
+ROM
+L
+(b)
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Time [s]
+-0.05
+0
+0.05
+IR [Pa]
+ROM
+FOM
+(c)
+102
+103
+Frequency (Hz)
+-40
+-20
+0
+20
+40
+SLP [dB]
+FOM
+ROM
+L
+(d)
+FIG. 7. Simulated pressure using the 3D FOM and ROM. a)
+CB domain sound pressure with Nrb = 275, b) CB domain
+frequency response with Nrb = 275, c) GR domain sound
+pressure with Nrb = 478, d) GR domain frequency response
+with Nrb = 478.
+TABLE V. Online stage boundary parameters for the 3D
+rooms. Values marked with * denotes parameterization.
+CB domain
+CE
+FL WE WS WW WN
+σmat [kNsm−4]
+12*
+-
+-
+7
+10*
+30
+dmat [m]
+0.05
+-
+-
+0.02
+0.05
+0.1
+d0 [m]
+0.06*
+-
+-
+0
+0
+0.1*
+Zs [kNsm−3]
+-
+50
+50
+-
+-
+50
+GR domain
+CE
+FL WE WS WW WN
+σmat [kNsm−4]
+10.5*
+-
+-
+7
+5.5*
+30
+dmat [m]
+0.05
+-
+-
+0.02
+0.05
+0.1
+d0 [m]
+0.025*
+-
+-
+0
+0
+0.03*
+Zs [kNsm−3]
+-
+50
+50
+-
+-
+50
+102
+103
+Frequency [Hz]
+-15
+-10
+-5
+0
+5
+10
+15
+L [dB]
+Nrb=81
+Nrb=275
+Nrb=543
+Nrb=837
+(a)
+102
+103
+Frequency [Hz]
+-15
+-10
+-5
+0
+5
+10
+15
+L [dB]
+Nrb=303
+Nrb=649
+Nrb=1500
+(b)
+FIG. 8. ∆L error. a) CB case 1 for Nrb = 81, 275, 543, 837.
+b) GR for Nrb = 303, 649, 1500.
+predict, especially when including a large number of pa-
+rameters in a complex scene. This study analyzes the be-
+haviour of realistic room acoustic scenarios by varying the
+geometry, source location, and inhomogeneous boundary
+in 2D and 3D. According to the singular energy decay,
+the geometry of the room does not have a significant im-
+pact on choosing the reduced basis and, therefore, the
+speedup. Moreover, the performance of the ROM when
+adding new parameters can also be estimated based on
+previous calculations, as it has been shown in Figure 4
+that the singular value decay is practically the same in
+8
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+
+ROMFOM0
+100
+200
+300
+400
+500
+600
+700
+10-6
+10-5
+10-4
+10-3
+10-2
+GR1.7kHz
+CB2.8kHz
+CB1.7kHz
+FIG. 9. Number of basis functions Nrb against the tolerance
+ǫP OD introduced in (9) for GR and CB at two different upper
+frequencies 1.7 kHz and 2.8 kHz
+the first basis functions when adding new parameters to
+the ROM, which is a clear indication of the potential of
+ROM for the type of applications studied.
+When increasing the dimension of a space (from 2D
+to 3D), a higher speedup is resulted for the same er-
+ror values (Figure 6). Speedups of up to two orders of
+magnitude are found for 2D and up to three orders for
+3D simulations when compared to FOM. These results
+agree with previous studies with homogeneous boundary
+conditions40.
+A recent MOR in time domain room acoustic sim-
+ulations using an automated Krylov subspace algorithm
+reported a speedup of 11–36 without introducing audi-
+ble differences for a simple scenario39. The present study
+shows higher speedups for a larger domain (considering
+the higher frequency). For the 2D domain (cases 1) with
+fu = 2 kHz, a reduction of the degree of freedom from
+DOF= 12039 to Nrb = 158 and Nrb = 752 resulted in a
+speedup of 142 (ǫrel = 0.35%) and 9 (ǫrel = 3.2×10−3%)
+respectively. Moreover, for the 3D cubic case, a reduction
+of the degree of freedom from DOF= 35937 to Nrb = 81
+and Nrb = 837 resulted in a speedup of 800 (ǫrel = 1.7%)
+and 47 (ǫrel = 5.8 × 10−3%) respectively.
+The difference in the reverberation between FOM
+and ROM has been quantified in relation to 5% JND of
+RT defined in54. Note that different studies show that the
+perception of the reverberation varies depending on the
+sound decay55 and the nature of the stimuli56–59, where
+the JND range from 3%-20%.
+V. CONCLUSION
+This study is concerned with the ability of ROM for
+use with different boundary conditions under the varia-
+tion of a considerable number of parameters and an inho-
+mogeneous distribution of absorption across the different
+surfaces of a room. First, our results confirm that the
+RBM is more favourable in terms of computational reduc-
+tion for symmetric problems, e.g., source positioned at
+63
+125
+250
+500
+1000
+2000
+Frequency [Hz]
+0
+100
+200
+300
+400
+500
+600
+T20 [ms]
+0.5
+0
+0
+0.5
+0.2
+0
+0.7
+0.1
+0
+0
+0.1
+FOMCB
+ROMCB
+FOMGR
+ROMGR
+(a)
+63
+125
+250
+500
+1000
+2000
+Frequency [Hz]
+0
+5
+10
+15
+20
+25
+30
+Number of JND
+Nrb=30
+Nrb=81
+Nrb=185
+Nrb=275
+Nrb=543
+Nrb=837
+(b)
+63
+125
+250
+500
+1000
+Frequency [Hz]
+0
+2
+4
+6
+8
+10
+Number of JND
+Nrb=62
+Nrb=303
+Nrb=649
+(c)
+FIG. 10.
+Comparison of reverberation time between FOM
+and ROM and their difference in terms of numbers of JND.
+a) Cube domain with Nrb = 185 (CB) and good ratio domain
+with Nrb = 649 (GR) including the number of JNDs per oc-
+tave band, b) Number of JNDs for CB, c) Number of JNDs
+for GR.
+the centre than in a corner. Second, results show that the
+singular value decay becomes more gentle when includ-
+ing more parameters into the ROM. Thirdly, speedups of
+one-two orders of magnitude are found for 2D, while two-
+three orders of magnitude are found for 3D. Fourthly, an
+analysis of reverberation time confirms that ROM pro-
+duces IRs of which the RTs are less than 5% from FOM.
+A smaller number of basis modes is needed in the trun-
+cated basis for the symmetric case in 3D, which shows a
+performance 365 times faster than the FOM.
+It can be concluded that complex ROMs with a large
+number of parameters and with acoustic materials dis-
+tributed inhomogeneously behave similarly to simple and
+homogeneous models and can achieve similar speedup
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+9
+
+performance. However, non-symmetric source positions
+and geometries and a large number of parameters can
+lead to a slower singular value decay, which may decrease
+the reduction if a large number of basis functions are in-
+cluded in the ROM. Although the FOM simulations are
+computationally costly, the price paid to create the ROM
+using FOM simulations is worthy for multiple design eval-
+uations, where different parameter configurations are to
+be explored or optimized for room acoustics.
+ACKNOWLEDGMENTS
+This research was done at the Technical University
+of Denmark and partly supported by Innovationsfonden,
+Denmark (Grant ID 9065-00115B), Rambøll Danmark
+A/S and Saint-Gobain Ecophon A/S, Sweden.
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+Appendix
+TABLE A1. Parameters selected to build the 2D ROM. Each
+row corresponds to a FOM simulation with the correspond-
+ing parameters marked for each column whose values are pre-
+sented in Table II.
+CEσ1 CEσ2 CEσ3 CEd1 CEd2 CEd3 CEd01 CEd02 CEd03 FLZ1 FLZ2 FLZ3 FLσ1 FLσ2 FLσ3 WLσ1 WLσ2 WLσ3 WRσ1 WRσ2 WRσ3
+FOM1
+X
+X
+X
+X
+X
+X
+FOM2
+X
+X
+X
+X
+X
+X
+FOM3
+X
+X
+X
+X
+X
+X
+FOM4
+X
+X
+X
+X
+X
+X
+FOM5
+X
+X
+X
+X
+X
+X
+FOM6
+X
+X
+X
+X
+X
+X
+FOM7
+X
+X
+X
+X
+X
+X
+FOM8
+X
+X
+X
+X
+X
+X
+FOM9
+X
+X
+X
+X
+X
+X
+FOM10
+X
+X
+X
+X
+X
+X
+FOM11
+X
+X
+X
+X
+X
+X
+FOM12
+X
+X
+X
+X
+X
+X
+FOM13
+X
+X
+X
+X
+X
+X
+FOM14
+X
+X
+X
+X
+X
+X
+FOM15
+X
+X
+X
+X
+X
+X
+FOM16
+X
+X
+X
+X
+X
+X
+FOM17
+X
+X
+X
+X
+X
+X
+FOM18
+X
+X
+X
+X
+X
+X
+FOM19
+X
+X
+X
+X
+X
+X
+FOM20
+X
+X
+X
+X
+X
+X
+FOM21
+X
+X
+X
+X
+X
+X
+TABLE A2. Parameters selected to build the 3D ROM. Each
+row corresponds to a FOM simulation with the correspond-
+ing parameters marked for each column whose values are pre-
+sented in Table III.
+CEσ1 CEσ2 CEσ3 CEd01 CEd02 CEd03 WWσ1 WWσ2 WWσ3 WNd01 WNd02 WNd03
+FOM1
+X
+X
+X
+X
+FOM2
+X
+X
+X
+X
+FOM3
+X
+X
+X
+X
+FOM4
+X
+X
+X
+X
+FOM5
+X
+X
+X
+X
+FOM6
+X
+X
+X
+X
+FOM7
+X
+X
+X
+X
+FOM8
+X
+X
+X
+X
+FOM9
+X
+X
+X
+X
+FOM10
+X
+X
+X
+X
+FOM11
+X
+X
+X
+X
+FOM12
+X
+X
+X
+X
+J. Acoust. Soc. Am. / 1 February 2023
+JASA/Sample JASA Article
+11
+

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+arXiv:2301.11797v1  [stat.ME]  27 Jan 2023
+From Classification Accuracy to Proper Scoring Rules:
+Elicitability of Probabilistic Top List Predictions
+Johannes Resin∗
+Heidelberg University
+Heidelberg Institute for Theoretical Studies
+January 30, 2023
+Abstract
+In the face of uncertainty, the need for probabilistic assessments has long been recognized in
+the literature on forecasting. In classification, however, comparative evaluation of classifiers
+often focuses on predictions specifying a single class through the use of simple accuracy
+measures, which disregard any probabilistic uncertainty quantification. I propose proba-
+bilistic top lists as a novel type of prediction in classification, which bridges the gap between
+single-class predictions and predictive distributions. The probabilistic top list functional is
+elicitable through the use of strictly consistent evaluation metrics. The proposed evalua-
+tion metrics are based on symmetric proper scoring rules and admit comparison of various
+types of predictions ranging from single-class point predictions to fully specified predictive
+distributions. The Brier score yields a metric that is particularly well suited for this kind of
+comparison.
+1
+Introduction
+In the face of uncertainty, predictions ought to quantify their level of confidence (Gneiting and
+Katzfuss, 2014). This has been recognized for decades in the literature on weather forecasting
+(Brier, 1950; Murphy, 1977) and probabilistic forecasting (Dawid, 1984; Gneiting and Raftery,
+2007). Ideally, a prediction specifies a probability distribution over potential outcomes. Such
+predictions are evaluated and compared by means of proper scoring rules, which quantify their
+value in a way that rewards truthful prediction (Gneiting and Raftery, 2007).
+In statistical
+classification and machine learning, the need for reliable uncertainty quantification has not gone
+unnoticed, as exemplified by the growing interest in the calibration of probabilistic classifiers
+(Guo et al., 2017; Vaicenavicius et al., 2019). However, classifier evaluation often focuses on the
+most likely class (i.e., the mode of the predictive distribution) through the use of classification
+accuracy and related metrics derived from the confusion matrix (Tharwat, 2020; Hui and Belkin,
+2021).
+Probabilistic classification separates the prediction task from decision making.
+This enables
+informed decisions that account for diverse cost-loss structures, for which decisions based simply
+∗This work has been supported by the Klaus Tschira Foundation. The author gratefully acknowledges financial
+support from the German Research Foundation (DFG) through grant number 502572912. The author would like
+to thank Timo Dimitriadis, Tobias Fissler, Tilmann Gneiting, Alexander Jordan, Sebastian Lerch and Fabian
+Ruoff for helpful comments and discussion.
+1
+
+on the most likely class may lead to adverse outcomes (Elkan, 2001; Gneiting, 2017). Probabilistic
+classification is a viable alternative to classification with reject option, where classifiers may refuse
+to predict a class if their confidence in a single class is not sufficient (Herbei and Wegkamp, 2006;
+Ni et al., 2019).
+In this paper, I propose probabilistic top lists as a way of producing probabilistic classifications
+in settings where specifying entire predictive distributions may be undesirable, impractical or
+even impossible. While multi-label classification serves as a key example of such a setting, the
+theory presented here applies to classification in general. I envision the probabilistic top list
+approach to be particularly useful in settings eluding traditional probabilistic forecasting, where
+the specification of probability distributions on the full set of classes is hindered by a large
+number of classes and missing (total) order. Consistent evaluation is achieved through the use
+of proper scoring rules.
+Whereas in traditional classification an instance is associated with a single class (e.g., cat or
+dog), multi-label classification problems (as reviewed by Tsoumakas and Katakis, 2007; Zhang
+and Zhou, 2014; Tarekegn et al., 2021) admit multiple labels for an instance (e.g., cat or dog or
+cat and dog).1 Applications of multi-label classification include text categorization (Zhang and
+Zhou, 2006), image recognition (Chen et al., 2019) and functional genomics (Barutcuoglu et al.,
+2006; Zhang and Zhou, 2006). Multi-label classification methods often output confidence scores
+for each label independently and the final label set prediction is determined by a simple cut-off
+(Zhang and Zhou, 2014). As this does not take into account label correlations, computing label set
+probabilities in a postprocessing step can improve predictions and probability estimates (Li et al.,
+2020) over simply multiplying probabilities to obtain label set probabilities. Probabilistic top lists
+offer a flexible approach to multi-label classification, which embraces the value of probabilistic
+information. In fact, the BR-rerank method introduced by Li et al. (2020) produces top list
+predictions. Yet, comparative performance evaluation focuses on (set) accuracy and the improper
+instance F1 score. This discrepancy has been a key motivation for this research.
+In probabilistic forecasting, a scoring rule assigns a numerical score to a predictive distribution
+based on the true outcome (Gneiting and Raftery, 2007). It is proper if the expected score is op-
+timized by the true distribution of the outcome of interest. Popular examples in classification are
+the Brier (or quadratic) score and the logarithmic (or cross entropy) loss (Gneiting and Raftery,
+2007; Hui and Belkin, 2021). When one is not interested in full predictive distributions, simple
+point predictions are frequently preferred. A meaningful point prediction admits interpretation
+in terms of a statistical functional (Gneiting, 2011). Point predictions are evaluated by means
+of consistent scoring or loss functions.
+Similar to proper scoring rules, a scoring function is
+consistent for a functional if the expected score is optimized by the true functional value of the
+underlying distribution. For example, accuracy (or, equivalently, misclassification or zero-one
+loss) is consistent for the mode in classification (Gneiting, 2017).
+Probabilistic top lists bridge the gap between mode forecasts and full predictive distributions in
+classification. In this paper, I define a probabilistic top-k list as a collection of k classes deemed
+most likely together with confidence scores quantifying the predictive probability associated
+with each of the k classes. The key question tackled in this work is how to evaluate such top list
+predictions in a consistent manner. To this end, I propose what I call padded symmetric scores,
+which are based on proper symmetric scoring rules. I show that the proposed padded symmetric
+scores are consistent for the probabilistic top-k list functional. The padded symmetric score of a
+probabilistic top list prediction is obtained from a symmetric proper scoring rule by padding the
+top list to obtain a fully specified distribution. The padded distribution divides the probability
+mass not accounted for by the top list’s confidence scores equally among the classes that are not
+included in the list. Padded symmetric scores exhibit an interesting property, which allows for
+1Multi-label classification is a special case of classification if classes are (re-)defined as subsets of labels.
+2
+
+balanced comparison of top lists of different length, as well as single-class point predictions and
+predictive distributions. Notably, the expected score of a correctly specified top list only depends
+on the top list itself and is invariant to other aspects of the true distribution. Comparability of
+top lists of differing length is ensured, as the expected score does not deteriorate upon increasing
+the length of the predicted top list. Nonetheless, if the scoring function is based on the Brier
+score, there is little incentive to provide unreasonably large top lists. In the case of a single-
+class prediction, the padded version of the Brier score reduces to twice the misclassification loss.
+Hence, the padded Brier score essentially generalizes classification accuracy.
+The remainder of the paper proceeds as follows. Section 2 recalls the traditional multi-class clas-
+sification problem with a focus on probabilistic classification and suitable evaluation metrics. A
+short introduction to the multi-label classification problem is also provided. Section 3 introduces
+probabilistic top lists and related notation and terminology used throughout this work. Section 4
+introduces some preliminary results on symmetric proper scoring rules and some results relating
+to the theory of majorization. These results are used in Section 5 to show that the padded sym-
+metric scores yield consistent scoring functions for the top list functionals. Section 6 discusses
+the comparison of various types of predictions using the padded Brier and logarithmic scores. A
+theoretical argument as well as numerical examples illustrate that the padded Brier score is well
+suited for this task. Section 7 concludes the paper.
+2
+Statistical Classification
+The top list functionals and the proposed scoring functions are motivated by multi-label classifi-
+cation, but they apply to other classification problems as well. Here, I give a short formal intro-
+duction to the general classification problem and related evaluation metrics from the perspective
+of probabilistic forecasting. In what follows, the symbol L refers to the law or distribution of a
+given random variable.
+2.1
+Traditional multi-class classification
+In the classical (multi-class) classification problem, one tries to predict the distinct class Y of an
+instance characterized by a vector of features X. Formally, the outcome Y is a random variable
+on a probability space (Ω, A, P) taking values in the set of classes Y of cardinality m ∈ N, and
+the feature vector X is a random vector taking values in some feature space X ⊆ Rd. Ideally, one
+learns the entire conditional distribution p(X) = L(Y | X) of Y given X through a probabilistic
+classifier c: X → P(Y) mapping the features of a given instance to a probability distribution
+from the set of probability distributions P(Y) on Y. The set P(Y) of probability distributions
+is typically identified with the probability simplex
+∆m−1 = {p ∈ [0, 1]m | p1 + · · · + pm = 1}
+by (arbitrarily) labeling the classes as 1, . . . , m, and probability distributions are represented by
+vectors p ∈ ∆m−1, where the i-th entry pi is the probability assigned to class i for i = 1, . . . , m.
+To ease notation in what follows, vectors in ∆m−1 are indexed directly by the classes in Y without
+explicit mention of any (re-)labeling.
+Proper scoring rules quantify the value of a probabilistic classification and facilitate comparison
+of multiple probabilistic classifiers (Gneiting and Raftery, 2007). A scoring rule is a mapping
+S: P(Y)×Y → R, which assigns a, possibly infinite, score S(p, y) from the extended real numbers
+R = R∪{±∞} to a predictive distribution p if the true class is y. Typically, scores are negatively
+3
+
+oriented in that lower scores are preferred. A scoring rule S is called proper if the true distribution
+p = L(Y ) of Y minimizes the expected score,
+E[S(p, Y )] ≤ E[S(q, Y )]
+for Y ∼ p and all p, q ∈ P(Y).
+(1)
+It is strictly proper if the inequality (1) is strict unless p = q. Prominent examples are the
+logarithmic score
+Slog(p, y) = − log py
+(2)
+and the Brier score
+SB(p, y) = (1 − py)2 +
+�
+z̸=y
+p2
+z = 1 − 2py +
+�
+z∈Y
+p2
+z.
+(3)
+Frequently, current practice does not focus on learning the full conditional distribution, but rather
+on simply predicting the most likely class, i.e., the mode of the conditional distribution p(X).
+This is formalized by a hard classifier c: X → Y aspiring to satisfy the functional relationship
+c(X) ∈ Mode(p(X)), where the mode functional is given by
+Mode(p) = arg max
+y∈Y
+py = {z ∈ Y | pz = max
+y∈Y py}
+(4)
+for p ∈ ∆m−1. Other functionals may be learned as well. When it comes to point forecasts of
+real-valued outcomes popular choices are the mean or a quantile, see for example Gneiting and
+Resin (2021). Formally, a statistical functional T: P(Y) → 2T reduces probability measures to
+certain facets in some space T . Note that the functional T maps a distribution to a subset in the
+power set 2T of T owing to the fact that the functional value may not be uniquely determined.
+For example, the mode (4) of a distribution is not unique if multiple classes are assigned the
+maximum probability. The probabilistic top lists introduced in Section 3 are a nonstandard
+example of a statistical functional, which lies at the heart of this work.
+Similar to the evaluation of probabilistic classifiers through the use of proper scoring rules, pre-
+dictions aimed at a statistical functional are evaluated by means of consistent scoring functions.
+Given a functional T, a scoring function is a mapping S: T ×Y → R, which assigns a score S(t, y)
+to a predicted facet t if the true class is y. A scoring function S is consistent for the functional
+T if the expected score is minimized by any prediction that is related to the true distribution of
+Y by the functional, i.e.,
+E[S(t, Y )] ≤ E[S(s, Y )]
+for Y ∼ p, t ∈ T(p) and all p ∈ P(Y), s ∈ T .
+(5)
+It is strictly consistent for T if the inequality (5) is strict unless s ∈ T(p). A functional T is
+called elicitable if a strictly consistent scoring function for T exists. For example, the mode (4)
+is elicited by the zero-one scoring function or misclassification loss (Gneiting, 2017)
+S(x, y) =
+1{x ̸= y},
+which is simply a negatively oriented version of the ubiquitous classification accuracy. As dis-
+cussed by Gneiting (2017) and references therein, decisions based on the mode are suboptimal if
+the losses invoked by different misclassifications are not uniform, which is frequently the case.
+(Strictly) Proper scoring rules arise as a special case of (strictly) consistent scoring functions if
+T is the identity on P(Y). Furthermore, any consistent scoring function yields a proper scoring
+rule if predictive distributions are reduced by means of the respective functional first (Gneiting,
+2011, Theorem 3). On the other hand, a point prediction x ∈ Y can be assessed by means of
+4
+
+a scoring rule, as the classes can be embedded in the probability simplex by identifying a class
+y ∈ Y with the point mass δy ∈ P(Y) in y. For example, applying the Brier score to a class
+prediction in this way yields twice the misclassification loss, SB(x, y) = SB(δx, y) = 2 ·
+1{x ̸= y}.
+Naturally, the true conditional distributions are unknown in practice and expected scores are
+estimated by the mean score attained across all instances available for evaluation purposes.
+2.2
+Multi-label classification
+In multi-label classification problems, an instance may be assigned multiple (class) labels. Here,
+I frame this as a special case of multi-class classification instead of an entirely different problem.
+Let L be the set of labels and Y ⊆ 2L be the set of label sets, i.e., classes are subsets of labels.
+In this setting, it may be difficult to specify a sensible predictive distribution on Y even for mod-
+erately sized sets of labels L, since the number of classes may grow exponentially in the number
+of labels. Extant comparative evaluation practices in multi-label classification focus mainly on
+hard classifiers ignoring the need for uncertainty quantification through probabilistic assessments
+(e.g., Tsoumakas and Katakis, 2007; Zhang and Zhou, 2014; Li et al., 2020; Tarekegn et al., 2021)
+with the exception of Read et al. (2011), who also consider a sum of binary logarithmic losses to
+evaluate the confidence scores associated with individual labels.
+Classification accuracy is typically referred to as (sub-)set accuracy in multi-label classification.
+Other popular evaluation metrics typically quantify the overlap between the predicted label set
+and the true label set.
+For example, the comparative evaluation by Li et al. (2020) reports
+instance F1 scores in addition to set accuracy, where instance F1 of a single instance is defined
+as
+SF1(x, y) =
+2 �
+ℓ∈L
+1{ℓ ∈ x} 1{ℓ ∈ y}
+�
+ℓ∈L
+1{ℓ ∈ x} + �
+ℓ∈L
+1{ℓ ∈ y}.
+(and the overall score is simply the average across all instances as usual). Note that this is a pos-
+itively oriented measure, i.e., higher instance F1 scores are preferred. Caution is advised, as the
+instance F1 score is not consistent for the mode as illustrated by the following example. Hence,
+evaluating the same predictions using set accuracy and instance F1 seems to be a questionable
+practice.
+Example 2.1. Let the label set L = {1, 2, 3, 4, 5} consist of five labels and the set of classes
+Y = 2L be the power set of the label set L. Consider the distribution p ∈ P(Y) that assigns all
+probability mass to four label sets as follows:
+p{1,2} = 0.28,
+p{1,3} = 0.24,
+p{1,4} = 0.24,
+p{1,5} = 0.24.
+Then the expected instance F1 score of the most likely label set {1, 2},
+E[SF1({1, 2}, Y )] = 0.64,
+given Y ∼ p is surpassed by predicting only the single label {1},
+E[SF1({1}, Y )] = 2
+3.
+3
+Probabilistic Top Lists
+In what follows, I develop a theory informing principled evaluation of top list predictions based
+on proper scoring rules. To this end, a concise mathematical definition of probabilistic top lists
+is fundamental.
+5
+
+Let k ∈ {0, . . . , m} be fixed. A (probabilistic) top-k list is a collection t = ( ˆY , ˆt) of a set ˆY ⊂ Y
+of k = | ˆY | classes together with a vector ˆt = (ˆty)y∈ ˆY ∈ [0, 1]k of confidence scores (or predicted
+probabilities) indexed by the set ˆY whose sum does not exceed one, i.e., �
+y∈ ˆY ˆty ≤ 1, and equals
+one if k = m. Let Tk denote the set of probabilistic top-k lists. On the one hand, the above
+definition includes the empty top-0 list t∅ = (∅, ()) for technical reasons. At the other extreme,
+top-m lists specify entire probability distributions on Y, i.e., Tm ≡ P(Y). The proxy probability
+π(t) :=
+1 − �
+y∈ ˆY ˆty
+m − k
+associated with a top-k list t = ( ˆY , ˆt) ∈ Tk of size k < m is the probability mass not accounted
+for by the top list t divided by the number of classes not listed. For a top-m list t ∈ Tm, the proxy
+probability π(t) ≡ 0 is defined to be zero. The padded probability distribution ˜t = (˜ty)y∈Y ∈ ∆m−1
+associated with a probabilistic top-k list t = ( ˆY , ˆt) ∈ Tk assigns the proxy probability π(t) to all
+classes not in ˆY , i.e.,
+˜ty =
+�
+ˆty,
+if y ∈ ˆY ,
+π(t),
+if y /∈ ˆY
+(6)
+for y ∈ Y.
+A top-k list t = ( ˆY , ˆt) is calibrated relative to a distribution p = (py)y∈Y ∈ ∆m−1 if the confidence
+score ˆty of class y matches the true class probability py for all y ∈ ˆY . A top-k list t = ( ˆY , ˆt) is true
+relative to a distribution p ∈ P(Y) if it is calibrated relative to p and ˆY consists of k most likely
+classes. There may be multiple true top-k lists for a given k ∈ N if the class probabilities are not
+distinct (i.e., some classes have the same probability). References to the true distribution of the
+outcome Y are usually omitted in what follows. For example, a calibrated top list is understood
+to be calibrated relative to the distribution L(Y ) of Y . The (probabilistic) top-k list functional
+Tk : P(Y) → Tk maps any probability distribution p ∈ P(Y) to the set
+Tk(p) =
+
+
+( ˆY , (py)y∈ ˆY ) ∈ Tk
+������
+ˆY ∈ arg max
+S⊂Y:|S|=k
+�
+y∈S
+py
+
+
+
+of top-k lists that are true relative to p. The top-m list functional Tm identifies P(Y) with Tm.
+A top-k list t ∈ Tk is valid if it is true relative to some probability distribution, i.e., there exists
+a distribution p ∈ P(Y) such that t ∈ Tk(p). Equivalently, a top-k list t = ( ˆY , ˆt) is valid if the
+associated proxy probability does not exceed the least confidence score, i.e., miny∈ ˆY ˆty ≥ π(t).
+Let ˜Tk ⊂ Tk denote the set of valid top-k lists. The following is a simple example illustrating the
+previous definitions.
+Example 3.1. Let k = 2, m = 4, Y = {1, 2, 3, 4} and Y ∼ p = (0.5, 0.2, 0.2, 0.1), i.e., P(Y = y) =
+py. There are two true top-2 lists, namely, T2(p) = {({1, 2}, (0.5, 0.2)), ({1, 3}, (0.5, 0.2))}. The
+list s = ({1, 4}, (0.5, 0.1)) is calibrated (relative to p), but fails to be valid, because it cannot be
+true relative to a probability distribution on Y. On the other hand, the list r = ({1, 4}, (0.5, 0.2))
+is valid, as it is true relative to q = (0.5, 0.2, 0.1, 0.2), but fails to be calibrated.
+An invalid top-k list t = ( ˆY , ˆt) contains a largest valid sublist t′ = ( ˆY ′, (ˆty)y∈ ˆY ′). The largest
+valid sublist is uniquely determined by recursively removing the class z ∈ arg miny∈ ˆY ˆty with the
+lowest confidence score from the invalid list until a valid list remains. Removing a class x ∈ ˆY
+with π(t) > ˆtx cannot result in a valid top list t′ = ( ˆY \ {x}, (ˆty)y∈ ˆY \{x}) as long as there is
+6
+
+another class z such that ˆtx ≥ ˆtz, because π(t) > π(t′) > ˆtx ≥ ˆtz. Similarly, removing a class
+x ∈ ˆY with π(t) ≤ ˆtx cannot prevent the removal of a class z if π(t) > ˆtz, because it does not
+decrease the proxy probability, π(t′) ≥ p(t). Hence, no sublist containing a class with minimal
+confidence score in the original list is valid and removal results in a superlist of the largest valid
+sublist.
+In what follows, I show how to construct consistent scoring functions for the top-k list functional
+using proper scoring rules. Recall from Section 2.1 that a scoring function S: Tk × Y → R is
+consistent for the top list functional Tk if the expected score under any probability distribution
+p ∈ P(Y) is minimized by any true top-k lists t ∈ Tk(p), i.e.,
+E[S(t, Y )] ≤ E[S(s, Y )]
+holds for Y ∼ p and any s ∈ Tk. It is strictly consistent if the expected score is minimized only
+by the true top-k lists t ∈ Tk(p), i.e., the inequality is strict for s /∈ Tk(p). The functional Tk
+is elicitable if a strictly consistent scoring function for Tk exists. In what follows, such a scoring
+function is constructed, giving rise to the following theorem.
+Theorem 3.2. The top-k list functional Tk is elicitable.
+Proof. This is an immediate consequence of either Theorem 5.4 or 5.6.
+As the image of Tk is ˜Tk by definition, invalid top-k lists may be ruled out a priori and the domain
+of S may be restricted to ˜Tk ×Y in the above definitions. This is essentially a matter of taste and
+the question is whether predictions must be valid or whether this should merely be encouraged
+by the use of a consistent scoring function. Any scoring function that is consistent for valid top
+list predictions can be extended by assigning an infinite score to any invalid top list regardless
+of the observation. In a sense, this reconciles both points of view, as an invalid prediction could
+not outperform any arbitrary valid prediction, thereby disqualifying it in comparison. In what
+follows, I focus on the construction of consistent scoring functions for valid top lists at first and
+propose a way of extending such scoring functions to invalid top lists that is less daunting than
+simply assigning an infinite score.
+4
+Mathematical Preliminaries
+This section introduces some preliminary results, which are used heavily in the next section.
+4.1
+Symmetric scoring rules
+The proposed scoring functions are based on symmetric proper scoring rules. Recall from Gneit-
+ing and Raftery (2007) that (subject to mild regularity conditions) any proper scoring rule
+S: P(Y) → R admits a Savage representation,
+S(p, y) = G(p) − ⟨G′(p), p⟩ + G′
+y(p),
+(7)
+in terms of a concave function G: ∆m−1 → R and a supergradient G′ : ∆m−1 → Rm of G, i.e., a
+function satisfying the supergradient inequality
+G(q) ≤ G(p) + ⟨G′(p), q − p⟩
+(8)
+for all p, q ∈ ∆m−1. Conversely, any function of the form (7) is a proper scoring rule. The function
+G is strictly concave if, and only if, S is strictly proper. It is called the entropy (function) of
+7
+
+S, and it is simply the expected score G(p) = E[S(p, Y )] under the posited distribution, Y ∼ p.
+The supergradient inequality (8) is strict if G is strictly concave and p ̸= q (Jungnickel, 2015,
+Satz 5.1.12).
+Let Sym(Y) denote the symmetric group on Y, i.e., the set of all permutations of Y. A scoring
+rule is called symmetric if scores are invariant under permutation of classes, i.e.,
+S((py), y) = S((pτ −1(y)), τ(y))
+holds for any permutation τ ∈ Sym(Y) and all y ∈ Y, p ∈ P(Y). Clearly, the entropy function
+G of a symmetric scoring rule is also symmetric, i.e., invariant to permutation in the sense that
+G(p) = G((pτ(y))) holds for any permutation τ ∈ Sym(Y) and any distribution p ∈ P(Y). Vice
+versa, any symmetric entropy function admits a symmetric proper scoring rule.
+Proposition 4.1. Let G: P(Y) → P(Y) be a concave symmetric function. Then there exists a
+supergradient G′ such that the Savage representation (7) yields a symmetric proper scoring rule.
+Proof. Let ¯G′ be a supergradient of G.
+Using the shorthand vτ = (vτ −1(y))y∈Y for vectors
+v = (vy)y∈Y ∈ Rm indexed by Y and permutations τ ∈ Sym(Y), define G′ by
+G′(p) =
+1
+| Sym(Y)|
+�
+τ∈Sym(Y)
+¯G′
+τ −1(pτ)
+for p ∈ P(Y). By symmetry of G and the supergradient inequality,
+G(q) = G(qτ) ≤ G(pτ) + ⟨ ¯G′(pτ), qτ − pτ⟩ = G(p) + ⟨ ¯G′
+τ −1(pτ), q − p⟩
+holds for all p, q ∈ P(Y) and τ ∈ Sym(Y). Summation over all τ ∈ Sym(Y) and division by the
+cardinality of the symmetric group Sym(Y) yields
+G(q) ≤
+1
+| Sym(Y)|
+�
+τ∈Sym(Y)
+(G(p) + ⟨ ¯G′
+τ −1(pτ), q − p⟩) = G(p) + ⟨G′(p), q − p⟩
+for any p, q ∈ P(Y). Therefore, G′ is a supergradient and the Savage representation (7) yields a
+symmetric scoring rule, since
+G′(p) =
+1
+| Sym(Y)|
+�
+τ∈Sym(Y)
+¯G′
+τ −1(pτ) =
+1
+| Sym(Y)|
+�
+τ∈Sym(Y)
+¯G′
+(τ◦ρ)−1(pτ◦ρ)
+=
+1
+| Sym(Y)|
+�
+τ∈Sym(Y)
+¯G′
+ρ−1◦τ −1(pτ◦ρ) =
+1
+| Sym(Y)|
+�
+τ∈Sym(Y)
+( ¯G′
+τ −1(pτ◦ρ))ρ−1
+=
+
+
+1
+| Sym(Y)|
+�
+τ∈Sym(Y)
+¯G′
+τ −1((pρ)τ)
+
+
+ρ−1
+= G′
+ρ−1(pρ)
+and
+⟨G′(p), p⟩ = ⟨G′
+ρ−1(pρ), p⟩ = ⟨G′(pρ), pρ⟩
+holds for any permutation ρ ∈ Sym(Y) and all p ∈ P(Y).
+On the other hand, not all proper scoring rules with symmetric entropy function are symmetric.
+The following result provides a necessary condition satisfied by supergradients of symmetric
+proper scoring rules.
+8
+
+Lemma 4.2. Let S be a symmetric proper scoring rule. If p ∈ ∆m−1 satisfies py = pz for y, z ∈
+Y, then the supergradient G′(p) at p in the Savage representation (7) satisfies G′
+y(p) = G′
+z(p).
+Proof. Let τ = (y z) be the permutation swapping y and z while keeping all other classes fixed.
+Using notation as in the proof of Proposition 4.1, the equality S(p, y) = S(pτ, τ(y)) holds by
+symmetry of S. Since p = pτ, the Savage representation (7) yields G′
+y(p) = G′
+τ(y)(p) = G′
+z(p).
+The Brier score (3) and the logarithmic score (2) are both symmetric scoring rules. The entropy
+function of the Brier score is given by
+G(p) = 1 −
+�
+y∈Y
+p2
+y,
+(9)
+whereas the entropy of the logarithmic score is given by
+G(p) = −
+�
+y∈Y
+py log(py)
+(see Gneiting and Raftery, 2007).
+4.2
+Majorization and Schur-concavity
+In this section, I adopt some definitions and results on majorization and Schur-concavity from
+Marshall et al. (2011). The theory of majorization is essentially a theory of inequalities, which
+covers many classical results and a plethora of mathematical applications not only in stochastics.
+For a vector v ∈ Rm, the vector v[ ] := (v[i])m
+i=1, where
+v[1] ≥ · · · ≥ v[m]
+denote the components of v in decreasing order, is called the decreasing rearrangement of v. A
+vector w ∈ Rm is a permutation of v ∈ Rm (i.e., w is obtained by permuting the entries of v)
+precisely if v[ ] = w[ ]. For vectors v, w ∈ Rm with equal sum of components, �
+i vi = �
+i wi, the
+vector v is said to majorize w, or v ≻ w for short, if the inequality
+k
+�
+i=1
+v[i] ≥
+k
+�
+i=1
+w[i]
+holds for all k = 1, . . . , m − 1.
+Let D ⊆ Rm. A function f : D → R is Schur-concave on D if v ≻ w implies f(v) ≤ f(w) for all
+v, w ∈ D. A Schur-concave function f is strictly Schur-concave if f(v) < f(w) holds whenever
+v ≻ w and v[ ] ̸= w[ ]. In particular, any symmetric concave function is Schur-concave and strictly
+Schur-concave if it is strictly concave (Marshall et al., 2011, Chapter 3, Proposition C.2 and
+C.2.c). Hence, the following lemma holds.
+Lemma 4.3. The entropy function of any symmetric proper scoring rule is Schur-concave. It
+is strictly Schur-concave if the scoring rule is strictly proper.
+A set D ⊂ Rm is called symmetric if v ∈ D implies w ∈ D for all vectors w ∈ Rm such that
+v[ ] = w[ ]. By the Schur-Ostrowski criterion (Marshall et al., 2011, Chapter 3, Theorem A.4
+and A.4.a) a continuously differentiable function f : D → R on a symmetric convex set D with
+non-empty interior is Schur-concave if, and only if, f is symmetric and the partial derivatives
+f(i)(v) =
+∂
+∂vi f(v) increase as the components vi of v decrease, i.e., f(i)(v) ≤ f(j)(v) if (and only
+if) vi ≥ vj.
+Unfortunately, this does not hold for supergradients of concave functions. The following is a
+slightly weaker condition, which applies to supergradients of symmetric concave functions.
+9
+
+Lemma 4.4 (Schur-Ostrowski condition for concave functions). Let f : D → R be a symmetric
+concave function on a symmetric convex set D, v ∈ D and f ′(v) be a supergradient of f at v,
+i.e., a vector satisfying the supergradient inequality
+f(w) ≤ f(v) + ⟨f ′(v), w − v⟩
+(10)
+for all w ∈ D. Then vi > vj implies f ′
+i(v) ≤ f ′
+j(v).
+Proof. For i = 1, . . . , m, let ei = (1{i = j})m
+j=1 denote the i-th vector of the standard basis
+of Rm. Let v ∈ D be such that vi > vj for some indices i, j and let 0 < ε ≤ vi − vj. Define
+w = v − εei + εej. Then v ≻ w (by Marshall et al., 2011, Chapter 2, Theorem B.6), because w
+is obtained from v through a so called ‘T -transformation’ (see Marshall et al., 2011, p. 32), i.e.,
+wi = λvi + (1 − λ)vj and wj = λvj + (1 − λ)vi with λ = vi−vj−ε
+vi−vj . By Schur-concavity of f, this
+implies f(v) ≤ f(w) and the supergradient inequality (10) yields
+ε(f ′
+j(v) − f ′
+i(v)) = ⟨f ′(v), w − v⟩ ≥ f(w) − f(v) ≥ 0.
+Hence, the inequality f ′
+j(v) ≥ f ′
+i(v) holds.
+With this, there is no need to restrict attention to differentiable entropy functions when applying
+the Schur-Ostrowski condition in what follows. Furthermore, true top-k lists can be characterized
+using majorization.
+Lemma 4.5. Let Y ∼ p be distributed according to p ∈ P(Y).
+The padded distribution ˜t
+associated with a true top-k list t ∈ Tk(p) majorizes the padded distribution ˜s associated with
+any calibrated top-k list s ∈ Tk.
+Proof. The sum of confidence scores �k
+i=1 ˜t[i] = �k
+i=1 p[i] ≥ �k
+i=1 ˜s[i] of a true top-k list is
+maximal among calibrated top-k lists by definition. Hence, the confidence score ˆt[i] = ˜t[i] of the
+true top-k list t = ( ˆY , ˆt) matches the i-th largest class probability p[i] for i = 1, . . . , k. Therefore,
+the partial sums �ℓ
+i=1 ˜t[i] = �ℓ
+i=1 p[i] ≥ �ℓ
+i=1 ˜s[i] across the largest confidence scores are also
+maximal for ℓ = 1, . . . , k − 1. Furthermore, the proxy probability π(t) =
+1−�k
+i=1 ˜t[i]
+m−k
+associated
+with a true top-k list is minimal among calibrated top-k lists. Hence, the partial sums
+ℓ
+�
+i=1
+˜t[i] = 1 − (m − ℓ)π(t) ≥ 1 − (m − ℓ)π(s) =
+ℓ
+�
+i=1
+˜s[i]
+are maximal for ℓ > k .
+5
+Consistent Top List Scores
+Having reviewed the necessary preliminaries, this section shows that the proposed padded sym-
+metric scores constitute a family of consistent scoring functions for the probabilistic top list
+functionals. The padded symmetric scores are defined for valid top lists and can be extended to
+invalid top lists by scoring the largest valid sublist, which yields a consistent scoring function.
+Strict consistency is preserved by adding an additional penalty term to the score of an invalid
+prediction.
+10
+
+5.1
+Padded symmetric scores
+From now on, let S: P(Y) → R be a proper symmetric scoring rule with entropy function G.
+The scoring rule S is extended to valid top-k lists for k = 0, 1, . . ., m − 1 by setting
+S(t, y) := S(˜t, y)
+for y ∈ Y, t ∈ ˜Tk, where ˜t ∈ ∆m−1 is the padded distribution (6) associated with the top-k list
+t. I call the resulting score S: �m
+k=0 ˜Tk × Y → R a padded symmetric score. For example, the
+logarithmic score (2) yields the padded logarithmic score
+Slog(( ˆY , ˆt), y) =
+�
+− log(ˆty),
+if y ∈ ˆY ,
+log(m − k) − log(1 − �
+z∈ ˆY ˆtz),
+otherwise,
+whereas the Brier score (3) yields the padded Brier score
+SB(( ˆY , ˆt), y) = 1 +
+�
+z∈ ˆY
+ˆt2
+z + (1 − �
+z∈ ˆY ˆtz)2
+m − k
+− 2 ·
+�ˆty,
+if y ∈ ˆY ,
+1−�
+z∈ ˆ
+Y ˆtz
+m−k
+,
+otherwise.
+(11)
+The following example shows that padded symmetric scores should not be applied to invalid top
+lists without further considerations.
+Example 5.1. If a padded symmetric score based on a strictly proper scoring rule is used to
+evaluate the invalid top-2 list s in Example 3.1, it attains a lower expected score than a true top
+list t ∈ T2(p), because ˜s = p, whereas ˜t ̸= p. Hence, the score would fail to be consistent.
+The following lemma shows that the expected score of a calibrated top list is fully determined
+by the top list itself and does not depend on (further aspects of) the underlying distribution.
+Lemma 5.2. Let S be a padded symmetric score. If p ∈ P(Y) is the true distribution of Y ∼ p,
+and t is a calibrated valid top list, then the expected score of the top list t matches the entropy
+of the padded distribution ˜t,
+E[S(t, Y )] = G(˜t).
+Proof. Let t = ( ˆY , ˆt) ∈ ˜Tk(p). Assume w.l.o.g. k < m (the claim is trivial if k = m) and let
+z ∈ Y \ ˆY . By Lemma 4.2 the supergradient at ˜t satisfies G′
+y(˜t) = G′
+z(˜t) for all y /∈ ˆY . Hence,
+the Savage representation (7) of the underlying scoring rule yields
+E[S(t, Y )] = G(˜t) − ⟨G′(˜t), ˜t⟩ +
+�
+y∈Y
+pyG′
+y(˜t)
+= G(˜t) −
+�
+y∈ ˆY
+(py − ˆty)G′
+y(˜ty) −
+
+�
+y /∈ ˆY
+py − (m − k)π(t)
+
+ G′
+z(˜t) = G(˜t),
+because t is calibrated.
+Padded symmetric scores exhibit an interesting property that admits balanced comparison of top
+list predictions of varying length. A top list score S: �m
+k=0 ˜Tk ×Y → R exhibits the comparability
+property if the expected score does not deteriorate upon extending a true top list, i.e., for
+k = 0, 1, . . ., m − 1 and any distribution p ∈ P(Y) of Y ∼ p,
+E[S(tk+1, Y )] ≤ E[S(tk, Y )]
+(12)
+11
+
+holds for tk ∈ Tk(p) and tk+1 ∈ Tk+1(p). The following theorem shows that padded symmetric
+scores in fact exhibit the comparability property.
+I use the comparability property to show
+consistency of the individual padded symmetric top-k list scores S | ˜Tk×Y and to extend these
+scores to invalid top lists. Section 6 provides further discussion and some numerical insights.
+Theorem 5.3. Padded symmetric scores exhibit the comparability property.
+Proof. Let S be a padded symmetric score and G be the concave entropy function of the under-
+lying proper scoring rule. Let Y ∼ p be distributed according to some distribution p ∈ P(Y)
+and let tk = ( ˆYk, (py)y∈ ˆYk) be a calibrated valid top-k list for some k = 0, 1, . . . , m − 1, which
+is extended by a calibrated valid top-(k + 1) list tk+1 = ( ˆYk+1, (py)y∈ ˆYk+1) in the sense that
+ˆYk+1 = ˆYk ∪ {z} for some z ∈ Y. It is easy to verify that ˜tk+1 ≻ ˜tk, since pz ≥ π(tk) ≥ π(tk+1).
+Hence, the inequality G(˜tk+1) ≤ G(˜tk) holds by Schur-concavity of G (Lemma 4.3), which yields
+the desired inequality of expected scores by Lemma 5.2.
+Clearly, there exists a true top-(k + 1) list tk+1 ∈ Tk+1(p) extending a true top-k list tk ∈ Tk(p)
+in the above sense. By a symmetry argument all true top lists of a given length have the same
+expected score and hence S exhibits the comparability property.
+Note that the proof of Theorem 5.3 shows that (12) holds for any calibrated valid extension
+tk+1 of a calibrated valid top list tk and not only true top lists. I proceed to show that padded
+symmetric scores restricted to valid top-k lists are consistent for the top-k list functional.
+Theorem 5.4. Let k ∈ {0, 1, . . ., m} be fixed and S: �m
+ℓ=0 ˜Tℓ × Y → R be a padded symmetric
+score. Then the restriction S | ˜Tk×Y of the score S to the set of valid top-k lists ˜Tk is consistent
+for the top-k list functional Tk. It is strictly consistent if the underlying scoring rule S |P(Y)×Y
+is strictly proper.
+Proof. Let p = (py)y∈Y ∈ P(Y) be the true probability distribution of Y ∼ p. Clearly, all true
+top-k lists in Tk(p) attain the same expected score by symmetry of the underlying scoring rule.
+Let t = ( ˆY , (py)y∈ ˆY ) ∈ Tk(ˆp) be a true top-k list and s = ( ˆZ, (ˆsy)y∈ ˆZ) ∈ ˜Tk be an arbitrary valid
+top-k list. To show consistency of S | ˜Tk×Y, it suffices to show that the valid top-k list s does not
+attain a lower (i.e., better) expected score than the true top-k list t. Strict consistency follows if
+the expected score of any s /∈ Tk(p) is higher than that of the true top-k list t.
+First, consider s /∈ Tk(p) to be a calibrated top-k list, i.e., ˆsy = py for all y ∈ ˆZ. Since ˜t majorizes
+˜s by Lemma 4.5, the inequality
+E[S(t, Y )] = G(˜t) ≤ G(˜s) = E[S(s, Y )]
+holds by Schur-concavity of the entropy function G (Lemma 4.3) and Lemma 5.2. If the under-
+lying scoring rule is strictly proper, the entropy function is strictly (Schur-)concave, and hence
+the inequality is strict.
+Now, consider s to be an uncalibrated top-k list and let r = ( ˆZ, (py)y∈ ˆZ) be the respective
+calibrated top-k list on the same classes. The calibrated top-k list r may not be valid and cannot
+be scored if this is the case. However, its largest valid sublist r′ = ( ˆZ′, (py)y∈ ˆ
+Z′) with ˆZ′ ⊆ ˆZ
+12
+
+can be scored. Let z ∈ Y \ ˆZ. The difference in expected scores
+E[S(s, Y )] − E[S(r′, Y )]
+= G(˜s) − G(˜r′) − ⟨G′(˜s), ˜s⟩ + ⟨G′(˜r′), ˜r′⟩ +
+�
+y∈Y
+py(G′
+y(˜s) − G′
+y(˜r′))
+(by the Savage representation (7))
+≥ ⟨G′(˜r′) − G′(˜s), ˜r′⟩ +
+�
+y∈Y
+py(G′
+y(˜s) − G′
+y(˜r′))
+(by the supergradient inequality (8))
+=
+�
+y∈ ˆZ\ ˆZ′
+(py − π(r′))(G′
+y(˜s) − G′
+z(˜r′)) +
+�
+y∈Y\ ˆ
+Z
+(py − π(r′))(G′
+z(˜s) − G′
+z(˜r′))
+(by Lemma 4.2)
+=
+�
+y∈ ˆZ\ ˆZ′
+(py − π(r′))(G′
+y(˜s) − G′
+z(˜s))
+(as �
+y∈Y\ ˆZ(py − π(r′)) = − �
+y∈ ˆZ\ ˆZ′(py − π(r′)))
+is nonnegative by the fact that (py −π(r′)) ≤ 0 for y ∈ ˆZ \ ˆZ′ (since r′ is the largest valid sublist)
+and Lemma 4.4 (and Lemma 4.2 if ˆsy = π(s) for some y ∈ ˆZ \ ˆZ′).
+Let k′ = | ˆZ′|. Then, r′ scores no better than a true top-k′ list tk′ ∈ Tk′(p), which in turn scores
+no better than t by the comparability property. Therefore,
+E[S(s, Y )] ≥ E[S(r′, Y )] ≥ E[S(tk′, Y )] ≥ E[S(t, Y )]
+holds. If the underlying scoring function is strictly proper, the difference in expected scores
+E[S(s, Y )] − E[S(r′, Y )] above is strictly positive by strictness of the supergradient inequality
+(Jungnickel, 2015, Satz 5.1.12), and hence E[S(t, Y )] < E[S(s, Y )] holds in this case, which
+concludes the proof.
+5.2
+Penalized extensions of padded symmetric scores
+The comparability property can be used to extend a padded symmetric score S to invalid top
+lists in a consistent manner. To this end, recall that t′ denotes the largest valid sublist of a top
+list t = ( ˆY , ˆt) ∈ Tk. Assigning the score of the largest valid sublist to an invalid top-k list yields
+a consistent score by the comparability property. Strict consistency of the padded symmetric
+score S is preserved by adding a positive penalty term cinvalid > 0 to the score of the largest
+valid sublist in the case of an invalid top list prediction. I call the resulting score extension
+S: �m
+k=0 Tk × Y → R, which assigns the score
+S(t, y) = S(t′, y) + cinvalid
+(13)
+to an invalid top list t ∈ Tk \ ˜Tk for k = 1, 2, . . . , m − 1, a penalized extension of a padded
+symmetric score. The following example illustrates that the positive penalty is necessary to
+obtain a strictly consistent scoring function.
+Example 5.5. Consider a setting similar to that of Example 3.1 with Y ∼ p = (0.4, 0.2, 0.2, 0.2).
+The padded distribution associated with the largest valid sublist t′ = ({1}, (0.4)) of the invalid
+list t = ({1, 2}, (0.4, 0.1)) matches the true distribution, ˜t′ = p, and hence the expected score of
+t in (13) is minimal if cinvalid = 0.
+The following theorem summarizes the properties of the proposed score extension.
+Theorem 5.6. Let k ∈ {0, 1, . . ., m} be fixed and S: �m
+ℓ=0 Tℓ × Y → R be a penalized extension
+(13) of a padded symmetric score with penalty term cinvalid ≥ 0. Then the restriction S |Tk×Y of
+13
+
+the score S to the set of top-k lists Tk is consistent for the top-k list functional Tk. It is strictly
+consistent if the underlying scoring rule S |P(Y)×Y is strictly proper and the penalty term cinvalid
+is nonzero.
+Proof. In light of Theorem 5.4, it remains to show that an invalid top-k list attains a worse
+expected score than a true top-k list t ∈ Tk(p) under the true distribution p ∈ P(Y) of Y ∼ p.
+To this end, let s ∈ Tk be invalid. By construction of the penalized extension, the top list s is
+assigned the score of its largest valid sublist s′ plus the additional penalty cinvalid. By consistency
+of the padded symmetric score and the comparability property, the expected score of s′ cannot
+fall short of the expected score of t. Hence, S |Tk×Y is consistent for the top-k list functional. If
+a positive penalty cinvalid > 0 is added, the score extension is strictly consistent given a strictly
+consistent padded symmetric score.
+6
+Comparability
+The comparability property (12) ensures that additional information provided by an extended
+true top list does not adversely influence the expected score. The information gain is quantified
+by a reduction in entropy, which depends on the underlying scoring rule. Ideally, a top list score
+encourages the prediction of classes that account for a substantial portion of probability mass,
+while offering little incentive to provide unreasonably large top lists. In what follows, I argue
+that the padded Brier score satisfies this requirement.
+Let S be a padded symmetric score with entropy function G (of the underlying proper scoring
+rule). Furthermore, let 1 ≤ k < m and t = ( ˆY , (ˆty)y∈ ˆY ) be a top-k list that accounts for most
+of the probability mass. In particular, assume that the unaccounted probability α = α(t) =
+1 − �
+y∈ ˆY ˆty is less than the least confidence score but nonzero, i.e.,
+0 < α < min
+y∈ ˆY
+ˆty.
+(14)
+Let Q = Q(t) = {p ∈ P(Y) | t ∈ Tk(p)} be the set of all probability measures relative to which
+t is a true top-k list. Let p ∈ Q assign the remaining probability mass α to a single class. Then
+p majorizes any q ∈ Q, and the distribution p has the lowest entropy, i.e., G(p) = minq∈Q G(q),
+by Schur-concavity of the entropy function (Lemma 4.3). As the expected score of the top list
+t is invariant under distributions in Q by Lemma 5.2, the relative difference in expected scores
+between the true top list t and the true distribution q ∈ Q is bounded by the relative difference
+in expected scores between t and p,
+G(˜t) − G(q)
+G(q)
+≤ G(˜t) − G(p)
+G(p)
+.
+The upper bound can be simplified by bounding the entropy of p from below, as G(p) ≥ G((1 −
+α, α, 0, . . . , 0)) by Schur-concavity of G.
+If S = SB is the padded Brier score (11) with entropy (9), the lower bound reduces to G(p) ≥
+G((1 − α, α, 0, . . . , 0)) = 2(α − α2) > α, since α < 0.5 by assumption (14) and hence 2α2 < α.
+With this the relative difference in expected scores has a simple upper bound,
+G(˜t) − G(p)
+G(p)
+= α2 − απ(t)
+2(α − α2) < α2
+α = α.
+For the padded logarithmic score no such bound exists and the deviation of the expected top
+list score from the optimal score can be severe, as illustrated in the following numerical example.
+14
+
+Table 1: Expected padded Brier scores and expected padded logarithmic scores of various types
+of true predictions and multiple distributions discussed in Example 6.1. Relative score differences
+(in percent) with respect to the optimal scores are in brackets.
+E[S(·, Y )]
+p
+S
+Mode(p)
+T1(p)
+T2(p)
+p
+p(h)
+SB
+0.02 (1.01%)
+0.0199 (0.38%)
+0.0198 (0%)
+0.0198
+p(m)
+SB
+1 (70.59%)
+0.6875 (17.28%)
+0.5867 (0.08%)
+0.5862
+p(l)
+SB
+1.5 (88.87%)
+0.7969 (0.34%)
+0.7955 (0.16%)
+0.7942
+p(h)
+Slog
+∞
+0.0699 (24.75%)
+0.0560 (0%)
+0.0560
+p(m)
+Slog
+∞
+1.3863 (32.49%)
+1.0532 (0.66%)
+1.0463
+p(l)
+Slog
+∞
+1.6021 (0.45%)
+1.5984 (0.23%)
+1.5948
+The example sheds some light on the behavior of the (expected) padded symmetric scores and
+demonstrates that top lists of length k > 1 may provide valuable additional information over a
+simple mode prediction.
+Example 6.1. Suppose there are m = 5 classes labeled 1, 2, . . ., 5 and the true (conditional)
+distribution p = p(x) = L(Y | X = x) of Y (given a feature vector x ∈ X) is known. Table 1
+features expected padded Brier and logarithmic scores of various types of truthful predictions
+under several distributions, as well as relative differences with respect to the optimal score. The
+considered distributions
+p(h) = (0.99, 0.01, 0, 0, 0),
+p(m) = (0.5, 0.44, 0.03, 0.02, 0.01),
+p(l) = (0.25, 0.22, 0.2, 0.18, 0.15).
+exhibit varying degrees of predictability. Distribution p(h) exhibits high predictability in the sense
+that a single class can be predicted with high confidence. Distribution p(m) exhibits moderate
+predictability in that it is possible to narrow predictions down to a small subset of classes
+with high confidence, but getting the class exactly right is a matter of luck. Distribution p(l)
+exhibits low predictability in the sense that all classes may well realize.
+Predictions are of
+increasing information content.
+The first prediction is the true mode, i.e., a hard classifier
+without uncertainty quantification that predicts class 1 under all considered distributions. The
+hard mode is interpreted as assigning all probability mass to the predicted class. Scores are
+obtained by embedding the predicted class in the probability simplex or, equivalent, by scoring
+the top-1 list ({1}, 1). The second prediction is the true top-1 list ({1}, p1), i.e., the mode with
+uncertainty quantification. The third prediction is the true top-2 list ({1, 2}, (p1, p2)) and the
+final prediction is the true distribution p itself.
+By consistency of the padded symmetric scores, the true top-1 lists score better in expectation
+than the mode predictions and by the comparability property, the true top-2 lists score better
+than the top-1 lists, while the true distributions attain the optimal scores. The mode predictions
+perform significantly worse than the probabilistic predictions, which highlights the importance
+of truthful uncertainty quantification. Note that the log score assigns an infinite score in case
+of the true outcome being predicted as having zero probability, hence the mode prediction is
+assigned an infinite score with positive probability.
+The expected padded Brier score of the probabilistic top-1 list under the highly predictable
+distribution p(h) is not far from optimal, whereas the respective logarithmic score is inflated by
+15
+
+the discrepancies between the padded and true distributions, even though the top list accounts
+for most of the probability mass (α = 0.01).
+Deviations from the optimal scores are more
+pronounced under the logarithmic score in all considered cases.
+Under the distribution exhibiting moderate predictability, the top-2 list prediction is much more
+informative than the top-1 list prediction, which results in a significantly improved score that
+is not far from optimal. Under the distribution exhibiting low predictability, all probabilistic
+predictions perform well, as there is little information to be gained.
+Estimation of small probabilities is frequently hindered by finite sample size. The specification of
+top list predictions in conjunction with the padded Brier score circumvents this issue, as the Brier
+score is driven by absolute differences in probabilities, whereas the logarithmic score emphasizes
+relative differences in probabilities. In other words, the padded distribution is deemed a good
+approximation of the true distribution if the true top list accounts for most of the probability
+mass by the Brier score.
+In light of these considerations, I conclude that the padded Brier score is suitable for the com-
+parison of top list predictions of varying length.
+7
+Concluding Remarks
+In this paper, I argued for the use of evaluation metrics rewarding truthful probabilistic assess-
+ments in classification. To this end, I introduced the probabilistic top list functionals, which
+offer a flexible probabilistic framework for the general classification problem. Padded symmetric
+scores yield consistent scoring functions, which admit comparison of various types of predictions.
+The padded Brier score appears particularly suitable, as top lists accounting for most of the
+probability mass obtain an expected padded Brier score that is close to optimal.
+The entropy of a distribution is a measure of uncertainty or information content. Majorization
+provides a relation characterizing common decreases in entropy shared by all symmetric proper
+scoring rules. In particular, for two distributions p ∈ P(Y) and q ∈ P(Y), the entropy of the
+distribution p does not exceed the entropy of q, i.e., G(p) ≤ G(q), if p majorizes q. The inequality
+is strict if the scoring rule is strictly proper and q is not a permutation of p.
+Similar to probabilistic top-k lists, a probabilistic top-β list with β ∈ (0, 1) may be defined as a
+minimal top list accounting for a probability mass of at least β. However, the padded symmetric
+scores proposed in this paper are not consistent for the top-β list functional, and the question
+whether this functional is elicitable constitutes an open problem for future research.
+As a simple alternative to the symmetric padded scores proposed in this paper, top-k error (Yang
+and Koyejo, 2020) is also a consistent scoring function for the top-k list functional, however, it is
+not strictly consistent, as it does not evaluate the confidence scores. On a related note, strictly
+proper scoring rules are essentially top-k consistent surrogate losses in the sense of Yang and
+Koyejo (2020). The idea of a consistent surrogate loss is to find a loss function that is easier to
+optimize than the target accuracy measure such that the confidence scores optimize accuracy.
+However, confidence scores need not represent probabilities. In contrast, strictly proper scoring
+rules elicit probabilities. Essentially, strictly proper scoring rules are consistent surrogates for
+any loss or scoring function that is consistent for a statistical functional.
+Typically, classes cannot simply be averaged. Therefore, combining multiple class predictions
+may be difficult, as majority voting may result in a tie, while learning individual voting weights
+or a meta-learner requires training data (see Kotsiantis et al., 2006, Section 8.3 for a review of
+classifier combination techniques). Probabilistic top lists facilitate the combination of multiple
+predictions, as confidence scores can simply be averaged, which may be an easy way to improve
+the prediction.
+16
+
+The prediction of probabilistic top lists appears particularly useful in problems, where classi-
+fication accuracy is not particularly high, as is frequently the case in multi-label classification.
+Probabilistic predictions are an informative alternative to classification with reject option. Fur-
+thermore, if it is possible to predict top lists of arbitrary length, the empty top-0 list may be
+seen as a reject option. Shifting focus towards probabilistic predictions may well increase pre-
+diction quality and usefulness in various decision problems, where misclassification losses are not
+uniform. The padded symmetric scores serve as general purpose evaluation metrics that account
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+

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+PLANAR EQUILIBRIUM MEASURE PROBLEM
+IN THE QUADRATIC FIELDS WITH A POINT CHARGE
+SUNG-SOO BYUN
+Abstract. We consider a two-dimensional equilibrium measure problem under the presence of quadratic po-
+tentials with a point charge and derive the explicit shape of the associated droplets. This particularly shows
+that the topology of the droplets reveals a phase transition: (i) in the post-critical case, the droplets are dou-
+bly connected domain; (ii) in the critical case, they contain two merging type singular boundary points; (iii)
+in the pre-critical case, they consist of two disconnected components. From the random matrix theory point
+of view, our results provide the limiting spectral distribution of the complex and symplectic elliptic Ginibre
+ensembles conditioned to have zero eigenvalues, which can also be interpreted as a non-Hermitian extension of
+the Marchenko-Pastur law.
+1. Introduction and Main results
+In this paper, we study a planar equilibrium problem with logarithmic interaction under the influence of
+quadratic potentials with a point charge. This problem is purely deterministic, but its motivation comes from
+the random world, more precisely, from the random matrix theory or the theory of two-dimensional Coulomb
+gases in general. To be more concrete, for given points ζ = (ζj)N
+j=1 ∈ CN of configurations, we consider the
+Hamiltonians
+HC
+N(ζ) :=
+�
+1≤j<k≤N
+log
+1
+|ζj − ζk|2 + N
+N
+�
+j=1
+W(ζj),
+(1.1)
+HH
+N(ζ) :=
+�
+1≤j<k≤N
+log
+1
+|ζj − ζk|2 +
+�
+1≤j≤k≤N
+log
+1
+|ζj − ¯ζk|2 + 2N
+N
+�
+j=1
+W(ζj).
+(1.2)
+Here W : C → R is a suitable function called the external potential. These are building blocks to define joint
+probability distributions
+(1.3)
+dPC
+N,β(ζ) ∝ e− β
+2 HC
+N(ζ)
+N
+�
+j=1
+dA(ζj),
+dPH
+N,β(ζ) ∝ e− β
+2 HH
+N(ζ)
+N
+�
+j=1
+dA(ζj),
+where dA(ζ) = d2ζ/π is the area measure and β > 0 is the inverse temperature. Both point processes PC
+N,β and
+PH
+N,β represent two-dimensional Coulomb gas ensembles [38, 55, 62]. In particular, if β = 2, they are also called
+determinantal and Pfaffian Coulomb gases respectively due to their special integrable structures, see [26, 28] for
+recent reviews on this topic. Furthermore, they have an interpretation as eigenvalues of non-Hermitian random
+matrices with unitary and symplectic symmetry. For instance, if W(ζ) = |ζ|2, the ensembles (1.3) corresponds
+to the eigenvalues of complex and symplectic Ginibre matrices [40].
+One of the fundamental questions regarding such point processes is their macroscopic/global behaviours as
+N → ∞. For the case β = 2, this can be regarded as a problem determining the limiting spectral distribution of
+given random matrices. The classical results in this direction include the circular law for the Ginibre ensembles.
+Date: January 3, 2023.
+Key words and phrases. Planar equilibrium measure problem, two-dimensional Coulomb gases, elliptic Ginibre ensemble, con-
+ditional point process, conformal mapping method, a non-Hermitian extension of the Marchenko-Pastur law.
+Sung-Soo Byun was partially supported by Samsung Science and Technology Foundation (SSTF-BA1401-51) and by the Na-
+tional Research Foundation of Korea (NRF-2019R1A5A1028324) and by a KIAS Individual Grant (SP083201) via the Center for
+Mathematical Challenges at Korea Institute for Advanced Study.
+1
+arXiv:2301.00324v1  [math-ph]  1 Jan 2023
+
+2
+SUNG-SOO BYUN
+As expected from the structure of the Hamiltonians (1.1) and (1.2), the macroscopic behaviours of the system
+can be effectively described using the logarithmic potential theory [59].
+For this purpose, let us briefly recap some basic notions in the logarithmic potential theory.
+Given a
+compactly supported probability measure µ on C, the weighted logarithmic energy IW [µ] associated with the
+potential W is given by
+(1.4)
+IW [µ] :=
+�
+C2 log
+1
+|z − w| dµ(z) dµ(w) +
+�
+C
+W dµ.
+For a general potential W satisfying suitable conditions, there exists a unique probability measure µW which
+minimises IW [µ]. Such a minimiser µW is called the equilibrium measure associated with W and its support
+SW := supp(µW ) is called the droplet. Furthermore, if W is C2-smooth in a neighbourhood of SW , it follows
+from Frostman’s theorem that µW is absolutely continuous with respect to dA and takes the form
+(1.5)
+dµW (z) = ∆W(z) · 1{z∈SW } dA(z),
+where ∆ := ∂ ¯∂ is the quarter of the usual Laplacian.
+In relation with the point processes (1.3), it is well known [31, 15, 41] that
+(1.6)
+µN,W := 1
+N
+N
+�
+j=1
+δζj → µW
+in the weak star sense of measure. From the statistical physics viewpoint, this convergence is quite natural since
+the weighted energy IW in (1.4) corresponds to the continuum limit of the discrete Hamiltonians (1.1) and (1.2)
+after proper renormalisations. (In the case of (1.2), it is required to further assume that W(ζ) = W(¯ζ).)
+Contrary to the density (1.5) of the measure µW , there is no general theory on the determination of its
+support SW . (See however [60] for a general theory on the regularity and [49] on the connectivity of the droplet
+associated with Hele-Shaw type potentials.) This leads to the following natural question.
+For a given potential W, what is the precise shape of the associated droplet?
+In view of the energy functional (1.4), this is a typical form of an inverse problem in the potential theory
+and is called an equilibrium measure problem.
+Beyond the case when W is radially symmetric (cf.
+[59,
+Section IV.6]), this problem is highly non-trivial even for some explicit potentials with a simple form, see
+[3, 12, 44, 14, 21, 35, 54, 36] for some recent works. Let us also stress that such a problem is important not
+only because it provides the intrinsic macroscopic behaviours of the point processes (1.3) but also because it
+plays the role of the first step to perform the Riemann-Hilbert analysis which gives rise to a more detailed
+statistical information (k-point functions) of the point processes, see [12, 13, 17, 18, 44, 45, 46, 47, 51, 53, 56]
+for extensive studies in this direction. In this work, we aim to contribute to the equilibrium problems associated
+with the potentials (1.7) and (1.14) below, which are of particular interest in the context of non-Hermitian
+random matrix theory.
+1.1. Main results. For given parameters τ ∈ [0, 1) and c ≥ 0, we consider the potential
+(1.7)
+Q(ζ) :=
+1
+1 − τ 2
+�
+|ζ|2 − τ Re ζ2�
+− 2c log |ζ|.
+When β = 2, the ensembles (1.3) associated with Q correspond to the distribution of random eigenvalues of the
+elliptic Ginibre matrices of size (c+1)N conditioned to have zero eigenvalues with multiplicity cN. We mention
+that such a model with c > 0 was also studied in the context of Quantum Chromodynamics [1].
+In (1.7), the logarithmic term can be interpreted as an insertion of a point charge, see [4, 33, 23, 22, 27]
+for recent investigations of the models (1.3) in this situation. Such insertion of a point charge has also been
+studied in the theory of planar orthogonal polynomials [12, 13, 17, 51, 52, 53, 16]. On the other hand, the
+parameter τ ∈ [0, 1) captures the non-Hermiticity of the model. To be more precise, the models (1.3) associated
+with Q interpolate the complex/symplectic Ginibre ensembles (τ = 0) with the Gaussian Unitary/Symplectic
+ensembles (τ = 1) conditioned to have zero eigenvalues, see Remark 1.6 for further discussion in relation to our
+main results.
+
+EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+3
+For the case c = 0, the terminology “elliptic” comes from the fact that the limiting spectrum Sτ,0 is given
+by the ellipse
+(1.8)
+Sτ,0 :=
+�
+(x, y) ∈ R2 :
+�
+x
+1 + τ
+�2
++
+�
+y
+1 − τ
+�2
+≤ 1
+�
+,
+which is known as the elliptic law, see e.g. [34, 37]. We refer to [50, 5, 8, 57, 20, 19] and references therein for
+more about the recent progress on the complex elliptic Ginibre ensembles and [43, 6, 24, 25] for their symplectic
+counterparts. For the rotationally invariant case when τ = 0, it is easy to show that the associated droplet S0,c
+is given by
+(1.9)
+S0,c :=
+�
+(x, y) ∈ R2 : c ≤ x2 + y2 ≤ 1 + c
+�
+,
+see e.g. [59, Section IV.6] and [26, Section 5.2].
+The primary goal of this work is to determine the precise shape of the droplet associated with the potential
+(1.7) for general τ ∈ [0, 1) and c ≥ 0. For this, we set some notations. Let us write
+(1.10)
+τc :=
+1
+1 + 2c
+for the critical non-Hermiticity parameter. For τ ∈ (τc, 1), we define
+(1.11)
+f(z) ≡ fτ(z) := (1 + τ)(1 + 2c)
+2
+(1 − az)(z − aτ)2
+z(z − a)
+,
+a = −
+1
+�
+τ(1 + 2c)
+.
+We are now ready to present our main result.
+Theorem 1.1. Let Q be given by (1.7). Then the droplet S ≡ Sτ,c = SQ of the equilibrium measure
+(1.12)
+dµQ(z) =
+1
+1 − τ 2 1S(z) dA(z)
+is given as follows.
+• (Post-critical case) If τ ∈ (0, τc], we have
+(1.13)
+Sτ,c =
+�
+(x, y) ∈ R2 :
+�
+x
+(1 + τ)√1 + c
+�2
++
+�
+y
+(1 − τ)√1 + c
+�2
+≤ 1 ,
+x2 + y2
+(1 − τ 2)c ≥ 1
+�
+.
+• (Pre-critical case) If τ ∈ [τc, 1), the droplet Sτ,c is the closure of the interior of the real analytic
+Jordan curves given by the image of the unit circle with respect to the map z �→ ±
+�
+f(z), where f is
+given by (1.11).
+(a) τ = 1/6 < τc
+(b) τ = 1/3 = τc
+(c) τ = 1/2 > τc
+Figure 1. The droplet Sτ,c, where c = 1 and a Fekete point configuration with N = 2048.
+
+2
+1
+0
+-1
+-2
+-2
+-1
+1
+-22
+1
+-1
+-2
+-2
+-1
+1
+-22
+1
+0
+-1
+-2
+-2
+-1
+1
+-24
+SUNG-SOO BYUN
+Note that if c = 0 (resp., τ = 0), the droplet (1.13) corresponds to (1.8) (resp., (1.9)). We mention that the
+post-critical case of Theorem 1.1 is indeed shown in a more general setup, see (2.1) and Proposition 2.1 below.
+Remark 1.2 (Phase transition of the droplet). In Theorem 1.1, we observe that if c > 0, the topology of
+the droplet reveals a phase transition. Namely, for the post-critical case when τ < τc, the droplet is a doubly
+connected domain, whereas for the pre-critical case τ > τc, it consists of two disconnected components. At
+criticality when τ = τc, the droplet contains two symmetric double points. We refer to [12, 14, 3, 36] for further
+models whose droplets reveal various phase transitions. Let us also mention that recently, there have been several
+works on the models (1.3) with multi-component droplets, see e.g. [13, 30, 9, 10]. In this pre-critical regime,
+some theta-function oscillations are expected to appear for various kinds of statistics; cf. [32, 9]. The precise
+asymptotic behaviours of the partition function would also be interesting in connection with the conjecture that
+these depend on the Euler index of the droplets, see [42, 29] and [26, Sections 4.1 and 5.3] for further discussion.
+Remark 1.3 (Fekete points and numerics). A configuration {ζj}N
+j=1 which makes the Hamiltonians (1.1) or
+(1.2) minimal is known as a Fekete configuration. This can be interpreted as the ensembles (1.3) with low
+temperature limit β = ∞, see e.g. [61, 58, 7, 11] and references therein. Since the droplet is independent of
+the inverse temperature β > 0 (excluding the high-temperature regime [2] when β = O(1/N)), the Fekete points
+are useful to numerically observe the shape of the droplets. In Figures 1 and 2, Fekete configurations associated
+with the Hamiltonian (1.1) are also presented, which show good fits with Theorems 1.1 and 1.4.
+Notice that the potential (1.7) and the droplet Sτ,c are invariant under the map ζ �→ −ζ. We now discuss an
+equivalent formulation of Theorem 1.1 under the removal of such symmetry. (See [36, Section 1.3] for a similar
+discussion in a vector equilibrium problem on a sphere with point charges.) The motivation for this formulation
+will be clear in the next subsection.
+For this purpose, we denote
+(1.14)
+�Q(ζ) :=
+2
+1 − τ 2
+�
+|ζ| − τ Re ζ
+�
+− 2c log |ζ|.
+By definition, the potentials Q in (1.7) and �Q in (1.14) are related as
+(1.15)
+Q(ζ) = 1
+2
+�Q(ζ2).
+Denoting by �S the droplet associated with �Q, it follows from [14, Lemma 1] that
+(1.16)
+S = {ζ ∈ C : ζ2 ∈ �S}.
+Due to the relation (1.16) and
+(1.17)
+∆ �Q(ζ) =
+1
+2(1 − τ 2)
+1
+|ζ|,
+we have the following equivalent formulation of Theorem 1.1.
+Theorem 1.4. Let �Q be given by (1.14). Then the droplet �S ≡ �Sτ,c = S �
+Q of the equilibrium measure
+(1.18)
+dµ �
+Q(ζ) =
+1
+2(1 − τ 2)
+1
+|ζ|1�S(ζ) dA(ζ)
+is given as follows.
+(i) (Post-critical case) If τ ∈ (0, τc], we have
+(1.19)
+�Sτ,c =
+�
+(x, y) ∈ R2 :
+� x − 2τ(1 + c)
+(1 + τ 2)(1 + c)
+�2
++
+�
+y
+(1 − τ 2)(1 + c)
+�2
+≤ 1 ,
+x2 + y2
+(1 − τ 2)2c2 ≥ 1
+�
+.
+(ii) (Pre-critical case) If τ ∈ [τc, 1), the droplet �Sτ,c is the closure of the interior of the real analytic
+Jordan curve given by the image of the unit circle with respect to the rational map z �→ f(z), where f
+is given by (1.11).
+
+EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+5
+(a) τ = 1/6 < τc
+(b) τ = 1/3 = τc
+(c) τ = 1/2 > τc
+Figure 2. The droplet �Sτ,c, where c = 1 and a Fekete point configuration with N = 2048.
+Remark 1.5 (Joukowsky transform in the critical case). If τ = τc with (1.10), we have a = 1/a = −1. Thus
+in this critical case, the rational function fτ in (1.11) is simplified as
+(1.20)
+fτc(z) = (1 + c)(z + τ)2
+z
+= (1 + c)
+�
+z + 2τ + τ 2
+z
+�
+.
+Note that compared to the general case (1.11), there is one less zero and one less pole in (1.20). Indeed, in the
+critical case, the rational map fτc is a (shifted) Joukowsky transform
+(1.21)
+fτc : Dc →
+�
+(x, y) ∈ R2 :
+� x − 2τ(1 + c)
+(1 + τ 2)(1 + c)
+�2
++
+�
+y
+(1 − τ 2)(1 + c)
+�2
+≥ 1
+�
+.
+In [3], a similar type of Joukowsky transform was used to solve an equilibrium measure problem. For the models
+under consideration in the present work, due to a more complicated form of the rational function (1.11), the
+required analysis for the associated equilibrium problem turns out to be more involved.
+Remark 1.6 (A non-Hermitian extension of the Marchenko-Pastur distribution). In the Hermitian limit τ ↑ 1,
+by (1.7) and (1.14), we have
+lim
+τ↑1 Q(x + iy) = V (x) :=
+�
+�
+�
+x2
+2 − 2c log |x|,
+if y = 0,
++∞
+otherwise,
+(1.22)
+lim
+τ↑1
+�Q(x + iy) = �V (x) :=
+�
+x − 2c log |x|,
+if y = 0, x > 0,
++∞
+otherwise.
+(1.23)
+Then the associated equilibrium measures are given by the well-known Marchenko-Pastur law (with squared
+variables) [38, Proposition 3.4.1], i.e.
+dµV (x) =
+1
+2π|x|
+�
+(λ2
++ − x2)(x2 − λ2
+−) · 1[−λ+,−λ−]∪[λ−,λ+] dx,
+(1.24)
+dµ�V (x) =
+1
+2πx
+�
+(λ2
++ − x)(x − λ2
+−) · 1[λ2
+−,λ2
++] dx,
+(1.25)
+where λ± := √2c + 1 ± 1, cf. Remark 2.3. Therefore one can interpret Theorem 1.1 (resp., Theorem 1.4) as
+a non-Hermitian generalisation of the Marchenko-Pastur distribution (1.24) (resp., (1.25)), see [8, Section 2]
+for more about the geometric meaning with the notion of the statistical cross-section. We also refer to [3] for
+another non-Hermitian extension of (1.24) and (1.25) in the context of the chiral Ginibre ensembles.
+
+3 
+2
+1
+0
+-1 
+2
+-3
+-2
+-1
+0
+1
+-2
+m
+43 
+2
+1
+0
+-1
+-2
+E-
+-2
+-1
+0
+1
+-2
+m
+43 
+2
+1
+0
+-1 
+-2
+E-
+-2
+I-
+0
+1
+-2
+m
+46
+SUNG-SOO BYUN
+Remark 1.7 (Inclusion relations of the droplets). Let us write
+S1 =
+�
+(x, y) ∈ R2 :
+�
+x
+1 + τ
+�2
++
+�
+y
+1 − τ
+�2
+≤ 1 + c
+�
+,
+S2 :=
+�
+(x, y) ∈ R2 : x2 + y2 ≤ (1 − τ 2)c
+�
+(1.26)
+and denote by �Sj (j = 1, 2) the image of Sj under the map z �→ z2. Then it follows from the definition (1.10)
+that
+(1.27)
+τ ∈ (0, τc)
+if and only if
+Sc
+1 ∩ S2 = ∅.
+By Theorems 1.1 and 1.4, for general τ ∈ [0, 1) and c ≥ 0, one can observe that
+(1.28)
+Sτ,c ⊆ S1 ∩ (Int S2)c,
+�Sτ,c ⊆ �S1 ∩ (Int �S2)c.
+Here, equality in (1.28) holds if and only if in the post-critical case. (This property holds in a more general
+setup, see Proposition 2.1.) On the other hand, in the pre-critical case one can interpret that the particles in
+Sc
+1 ∩ S2 are smeared out to S1 ∩ Sc
+2 which makes the inclusion relations (1.28) strictly hold, see Figure 3.
+(a) Sτ,c
+(b) �Sτ,c
+Figure 3. The droplets Sτ,c and �Sτ,c in the pre-critical case, where c = 2 and τ = 0.7 > τc.
+Here, the dashed lines display the boundaries of Sj and �Sj (j = 1, 2).
+1.2. Outline of the proof. Recall that µW is a unique minimiser of the energy (1.4). It is well known that
+the equilibrium measure µW is characterised by the variational conditions (see [59, p.27])
+�
+log
+1
+|ζ − z|2 dµW (z) + W(ζ) = C,
+q.e.
+if ζ ∈ SW ;
+(1.29)
+�
+log
+1
+|ζ − z|2 dµW (z) + W(ζ) ≥ C,
+q.e.
+if ζ /∈ SW .
+(1.30)
+Here, q.e. stands for quasi-everywhere. (Nevertheless, this notion is not important in the sequel as we will show
+that for the models we consider the conditions (1.29) and (1.30) indeed hold everywhere.)
+Due to the uniqueness of the equilibrium measure, all we need to show is that if W = Q, then µQ in
+(1.12) satisfies the variational principles (1.29) and (1.30). Equivalently, by (1.16), it also suffices to show the
+variational principles for the equilibrium measure µ �
+Q in (1.18).
+However, it is far from being obvious to obtain the “correct candidate” of the droplets. Perhaps one may
+think that at least for the post-critical case, the shape of the droplet (1.13) is quite natural given the well-known
+cases (1.8) and (1.9) as well as the fact that the area of Sτ,c should be (1 − τ 2)π. On the other hand, for the
+pre-critical case, one can easily notice that there is some secret behind deriving the explicit formula of the
+rational function (1.11). To derive the correct candidate, we use the conformal mapping method with the help
+of the Schwarz function, see Appendix A.
+Remark 1.8 (Removal of symmetry). We emphasise that the conformal mapping method does not work for
+the multi-component droplet, i.e. the pre-critical case of Theorem 1.1. This is essentially due to the lack of
+the Riemann mapping theorem. Nevertheless, one can observe that once we remove the symmetry ζ �→ −ζ, the
+droplet in the pre-critical case of Theorem 1.4 is simply connected. This explains the reason why we need the
+idea of removing symmetry.
+
+-
+1
+1
+-1
+1
+-EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+7
+The rest of this paper is organised as follows.
+• In Section 2, we prove Theorems 1.1 and 1.4.
+In Subsection 2.1, we show the post-critical case of
+Theorem 1.1 in a more general setup, see Proposition 2.1. On the other hand, in Subsection 2.2, we
+show the pre-critical case of Theorem 1.4. Then by the relation (1.16), these complete the proof of our
+main results.
+• This article contains two appendices.
+In Appendix A, we explain the conformal mapping method
+to derive the “correct candidate” of the droplets. In Appendix B, we present a way to solve a one-
+dimensional equilibrium problem in Remark 2.3, which shares a common feature with the conformal
+mapping method. These appendices are made only for instructive purposes and the readers who only
+want the proof of the main theorems may stop at the end of Section 2.
+2. Proof of main theorem
+In this section, we show Theorems 1.1 and 1.4.
+2.1. Post critical cases. Extending (1.7), we consider the potential
+(2.1)
+Qp(ζ) :=
+1
+1 − τ 2
+�
+|ζ|2 − τ Re ζ2�
+− 2c log |ζ − p|,
+p ∈ C.
+For the case τ = 0, the shape of the droplet associated with the potential (2.1) was fully characterised in [12].
+(In this case, it suffices to consider the case p ≥ 0 due to the rotational invariance.) In particular, it was shown
+that if
+(2.2)
+c < (1 − p2)2
+4p2
+,
+τ = 0,
+p ≥ 0,
+the droplet is given by S = D(0, √1 + c) \ D(p, √c), where D(p, R) is the disc with centre p and radius R, cf.
+see Remark A.5 for the other case c > (1 − p2)2/(4p2).
+To describe the droplets associated with Qp, we denote
+(2.3)
+S1 :=
+�
+(x, y) ∈ R2 :
+�
+x
+1 + τ
+�2
++
+�
+y
+1 − τ
+�2
+≤ 1 + c
+�
+and
+(2.4)
+S2 :=
+�
+(x, y) ∈ R2 : (x − Re p)2 + (y − Im p)2 ≤ (1 − τ 2)c
+�
+.
+Then we obtain the following.
+Proposition 2.1. Suppose that the parameters τ, c ∈ R and p ∈ C are given to satisfy
+(2.5)
+S2 ⊂ S1,
+where S1 and S2 are given by (2.3) and (2.4). Then the droplet S ≡ SQp associated with (2.1) is given by
+(2.6)
+S = S1 ∩ (Int S2)c.
+See Figure 4 for the shape of the droplets and numerical simulations of Fekete point configurations. We
+remark that with slight modifications, Proposition 2.1 can be further extended to the case with multiple point
+charges, i.e. the potential of the form
+(2.7)
+1
+1 − τ 2
+�
+|ζ|2 − τ Re ζ2�
+− 2
+�
+cj log |ζ − pj|,
+pj ∈ C,
+cj ≥ 0.
+(See Remark A.4 for a related discussion.) Let us also mention that a similar statement for an equilibrium
+problem on the sphere was shown in [21].
+For a treatment of a more general case, we refer the reader to
+[35, 54, 36].
+Remark 2.2. If p = 0, the condition (2.5) corresponds to
+(2.8)
+τ <
+1
+1 + 2c = τc.
+Therefore Proposition 2.1 for the special value p = 0 gives Theorem 1.1 (i).
+As a consequence, by (1.16),
+Theorem 1.4 (i) also follows. We also mention that if τ = 0 and p > 0, the condition (2.5) coincides with (2.2).
+
+8
+SUNG-SOO BYUN
+(a) p =
+2
+21
+√
+14 i
+(b) p = 2
+7
+√
+14
+(c) p = 3
+5 + 1
+5i
+Figure 4. The droplet S in Proposition 2.1, where τ = 1/3 and c = 1/7. Here, a Fekete point
+configuration with N = 2048 is also displayed.
+Remark 2.3 (Equilibrium measure in the Hermitian limit). Before moving on to the planar equilibrium problem
+for (2.1), we first discuss the one-dimensional problem arising in the Hermitian limit. For p ∈ R, the Hermitian
+limit τ ↑ 1 of the potential Qp is given by
+(2.9)
+lim
+τ↑1 Qp(x + iy) = Vp(x) :=
+�
+�
+�
+x2
+2 − 2c log |x − p|,
+if y = 0,
++∞
+otherwise.
+Then one can show that the associated equilibrium measure µV ≡ µVp is given by
+(2.10)
+dµV (x)
+dx
+=
+�
+− �4
+j=1(x − λj)
+2π|x − p|
+· 1[λ1,λ2]∪[λ3,λ4](x),
+where
+λ1 = p − 2 −
+�
+(p + 2)2 + 8c
+2
+,
+λ2 = p + 2 −
+�
+(p − 2)2 + 8c
+2
+,
+(2.11)
+λ3 = p − 2 +
+�
+(p + 2)2 + 8c
+2
+,
+λ4 = p + 2 +
+�
+(p − 2)2 + 8c
+2
+.
+(2.12)
+We remark that when p = 0, it recovers (1.24). See Figure 5 for the graphs of the equilibrium measure µVp. The
+equilibrium measure (2.10) follows from the standard method using the Stieltjes transform and the Sokhotski-
+Plemelj inversion formula. For reader’s convenience, we provide a proof of (2.10) in Appendix B.
+(a) p = 0
+(b) p = 1
+(c) p = 4
+Figure 5. Graphs of the equilibrium measure µVp, where c = 1.
+
+15
+10
+0.5 
+0.0 
+0.5
+1.0
+1.5
+-1.5
+-1.0 
+0.5 
+0'0
+0.5
+10
+1'5 15
+10
+0.5 
+0.0
+0.5
+1.0
+1.5
+1.5
+-1.0 
+0.5
+0'0
+0.5 
+10
+150.35
+0.30
+0.25
+0.20
+0.15
+0.10
+0.05
+2
+0
+2
+40.35
+0.30
+0.25
+0.20
+0.15
+0.10
+0.05
+-2
+0
+2
+40.35
+0.30
+0.25
+0.20
+0.15
+0.10
+0.05
+-2
+0
+2
+415
+10
+0.5 
+0.0
+0.5
+1.0
+1.5
+1.5
+-1.0 
+0.5
+0'0
+0.5 
+10
+15EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+9
+In the rest of this subsection, we prove Proposition 2.1. First, let us show the following elementary lemmas.
+Lemma 2.4. For a, b > 0, let
+K :=
+�
+(x, y) ∈ R2 :
+�x
+a
+�2
++
+�y
+b
+�2
+≤ 1
+�
+.
+Then we have
+(2.13)
+�
+K
+1
+ζ − z dA(z) =
+�
+�
+�
+�
+�
+¯ζ − a − b
+a + b ζ
+if ζ ∈ K,
+2ab
+a2 − b2
+�
+ζ −
+�
+ζ2 − a2 + b2
+�
+otherwise.
+In particular, for ζ ∈ K, there exists a constant c0 ∈ R such that
+(2.14)
+�
+K
+log |ζ − z|2 dA(z) = |ζ|2 − a − b
+a + b Re ζ2 + c0.
+Remark 2.5. The Cauchy transform in (2.13) is useful to explicitly compute the moments of the equilibrium
+measure. Namely, by definition, we have
+�
+K
+1
+ζ − z dA(z) = 1
+ζ
+∞
+�
+k=0
+1
+ζk
+�
+K
+zk dA(z),
+ζ → ∞.
+On the other hand, we have
+ζ −
+�
+ζ2 − a2 + b2 = a2 − b2
+ζ
+∞
+�
+k=0
+� 1/2
+k + 1
+�(b2 − a2)k
+ζ2k
+,
+ζ → ∞.
+Combining the above equations with (2.13), we obtain that for any non-negative integer k,
+(2.15)
+1
+ab
+�
+K
+z2k dA(z) = 2
+� 1/2
+k + 1
+�
+(b2 − a2)k.
+Proof of Lemma 2.4. Recall that D is the unit disc with centre the origin.
+Then the Joukowsky transform
+f : ¯Dc → Kc is given by
+(2.16)
+f(z) = a + b
+2
+z + a − b
+2
+1
+z .
+By applying Green’s formula, we have
+(2.17)
+�
+K
+1
+ζ − z dA(z) =
+1
+2πi
+�
+∂K
+¯z
+ζ − z dz + ¯ζ · 1{ζ∈K}.
+Furthermore, by the change of variable z = f(w), it follows that
+�
+∂K
+¯z
+ζ − z dz =
+�
+∂D
+f(1/ ¯w)
+ζ − f(w)f ′(w) dw =
+�
+∂D
+gζ(w) dw,
+(2.18)
+where gζ is the rational function given by
+(2.19)
+gζ(w) :=
+1
+ζ − f(w)
+�a + b
+2
+1
+w + a − b
+2
+w
+��a + b
+2
+− a − b
+2
+1
+w2
+�
+.
+Observe that
+ζ = f(w)
+if and only if
+w = w±
+ζ := ζ ±
+�
+ζ2 − a2 + b2
+a + b
+,
+i.e. the points w±
+ζ are solutions to the quadratic equation
+(a + b)w2 − 2ζ w + (a − b) = 0.
+Here, the branch of the square root is chosen such that
+w−
+ζ → 0
+ζ → ∞.
+By above observation, the function gζ has poles only at
+0,
+w+
+ζ ,
+w−
+ζ .
+
+10
+SUNG-SOO BYUN
+Moreover note that
+ζ ∈ K
+if and only if
+w±
+ζ ∈ D.
+Notice that if ζ ∈ Kc, then w−
+ζ ∈ D and w+
+ζ ∈ Dc.
+Using the residue calculus, we have
+(2.20)
+Res
+w=0
+�
+gζ(w)
+�
+= a + b
+a − b ζ.
+On the other hand, we have
+(2.21)
+Res
+w=w±
+ζ
+�
+gζ(w)
+�
+= −f(1/ ¯w±
+ζ ) = −a + b
+2
+1
+w±
+ζ
+− a − b
+2
+w±
+ζ .
+In particular,
+(2.22)
+Res
+w=w+
+ζ
+�
+gζ(w)
+�
++ Res
+w=w−
+ζ
+�
+gζ(w)
+�
+= −2a2 + b2
+a2 − b2 ζ.
+Combining all of the above, we obtain the desired identity (2.13). The second assertion immediately follows
+from (2.13) and the real-valuedness of ζ �→
+�
+K log |ζ − z|2 dA(z).
+□
+Lemma 2.6. For R > 0 and p ∈ C we have
+(2.23)
+�
+D(p,R)
+log |ζ − z| dA(z) =
+�
+�
+�
+R2 log |ζ − p|
+if ζ /∈ D(p, R),
+R2 log R − R2
+2 + |ζ − p|2
+2
+otherwise.
+Proof. First, recall the well-known Jensen’s formula: for r > 0,
+(2.24)
+1
+2π
+� 2π
+0
+log |ζ − reiθ| dθ =
+�
+log r
+if r > |ζ|,
+log |ζ|
+otherwise.
+By the change of variables, we have
+�
+D(p,R)
+log |ζ − z| dA(z) =
+�
+D(0,R)
+log |ζ − p − z| dA(z) = 1
+π
+� R
+0
+r
+� 2π
+0
+log |ζ − p − reiθ| dθ dr.
+Suppose that ζ /∈ D(p, R). Then by applying (2.24), we have
+1
+π
+� R
+0
+r
+� 2π
+0
+log |ζ − p − reiθ| dθ dr = 2
+� R
+0
+r log |ζ − p| dr = R2 log |ζ − p|.
+On the other hand if ζ ∈ D(p, R), we have
+1
+π
+� R
+0
+r
+� 2π
+0
+log |ζ − p − reiθ| dθ dr = 2
+� |ζ−p|
+0
+r log |ζ − p| dr + 2
+� R
+|ζ−p|
+r log r dr
+= R2 log R − R2
+2 + |ζ − p|2
+2
+,
+which completes the proof.
+□
+We are now ready to complete the proof of Proposition 2.1.
+Proof of Proposition 2.1. Note that by (1.5), the equilibrium measure µ associated with Qp is of the form
+(2.25)
+dµ(z) := ∆Qp(z) · 1S(z) dA(z) =
+1
+1 − τ 2 · 1S(z) dA(z).
+Due to the assumption (2.5), we have
+�
+log
+1
+|ζ − z|2 dµ(z) =
+1
+1 − τ 2
+� �
+S1
+log
+1
+|ζ − z|2 dA(z) −
+�
+S2
+log
+1
+|ζ − z|2 dA(z)
+�
+.
+
+EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+11
+Note that by Lemma 2.4, there exists a constant c0 such that
+(2.26)
+�
+S1
+log
+1
+|ζ − z|2 dA(z) = −|ζ|2 + τ Re ζ2 − c0.
+On the other hand, by Lemma 2.6, we have
+(2.27)
+�
+S2
+log
+1
+|ζ − z|2 dA(z) = −2(1 − τ 2)c log |ζ − p|.
+Combining (2.26), (2.27) and (2.1), we obtain
+(2.28)
+�
+log
+1
+|ζ − z|2 dµ(z) = −Qp(ζ) −
+c0
+1 − τ 2 ,
+which leads to (1.29).
+Next, we show the variational inequality (1.30). Note that if ζ ∈ S2, it immediately follows from Lemma 2.6.
+Thus it is enough to verify (1.30) for the case ζ ∈ Sc
+1. Let
+(2.29)
+Hp(ζ) :=
+�
+log
+1
+|ζ − z|2 dµ(z) + Qp(ζ).
+Suppose that the variational inequality (1.30) does not hold. Then since Hp(ζ) → ∞ as ζ → ∞, there exists
+ζ∗ ∈ Sc
+1 such that
+(2.30)
+∂ζHp(ζ)|ζ=ζ∗ = 0.
+On the other hand, by Lemmas 2.4 and 2.6, if ζ ∈ Sc
+1, the Cauchy transform of the measure µ is computed as
+(2.31)
+� dµ(z)
+ζ − z = 1
+2τ
+�
+ζ −
+�
+ζ2 − 4τ(1 + c)
+�
+−
+c
+ζ − p.
+Together with (2.1), this gives rise to
+∂ζQp(ζ) −
+� dµ(z)
+ζ − z =
+1
+1 − τ 2
+�
+¯ζ − τζ
+�
+− 1
+2τ
+�
+ζ −
+�
+ζ2 − 4τ(1 + c)
+�
+.
+(2.32)
+Then it follows that the condition ∂ζHp(ζ) = 0 is equivalent to
+(2.33)
+(1 + τ 2)|ζ|2 − τ(ζ2 + ¯ζ2) = (1 − τ 2)2(1 + c).
+Therefore, by (2.3), one can notice that ∂ζHp(ζ) = 0 if and only if ζ ∈ ∂S1. This yields a contradiction with
+the assumption ζ∗ ∈ Sc
+1. Therefore we conclude that the variational inequality (1.30) holds for ζ ∈ Sc, which
+completes the proof.
+□
+Remark 2.7. Let us denote by
+(2.34)
+mk :=
+�
+zk dµ(z)
+the k-th moment of the equilibrium measure. Notice that the Cauchy transform of µ satisfies the asymptotic
+expansion
+(2.35)
+� dµ(z)
+ζ − z = 1
+ζ
+∞
+�
+k=0
+mk
+ζk ,
+ζ → ∞.
+Using this property and (2.31), after straightforward computations, one can verify that the equilibrium measure
+µ in Proposition 2.1 has the moments
+(2.36)
+m2k = 2
+(2k − 1)!
+(k − 1)!(k + 1)!τ k(1 + c)k+1 − c p2k,
+m2k+1 = −c p2k+1.
+Notice in particular that if p = 0, all odd moments vanish.
+
+12
+SUNG-SOO BYUN
+2.2. Pre-critical case. In this subsection, we show Theorem 1.4 (ii). Then by (1.16), Theorem 1.1 (ii) follows.
+Proof of Theorem 1.4 (ii). Recall that �Q is given by (1.14) and that all we need to show is the variational
+principles (1.29) and (1.30) for W = �Q. For this, similar to above, let
+(2.37)
+H(ζ) :=
+�
+log
+1
+|ζ − z|2 d�µ(z) + �Q(ζ),
+where �µ is the equilibrium measure associated with �Q. Then
+(2.38)
+∂ζH(ζ) = ∂ζ �Q(ζ) − C(ζ) =
+1
+1 − τ 2
+��
+¯ζ
+ζ − τ
+�
+− c
+ζ − C(ζ),
+where C(ζ) is the Cauchy transform of �µ given by
+(2.39)
+C(ζ) =
+1
+2(1 − τ)2
+�
+�S
+1
+ζ − z
+1
+|z| dA(z).
+Here, we have used (1.5).
+Applying Green’s formula, we have
+(1 − τ 2)C(ζ) =
+1
+2πi
+�
+∂ �S
+1
+ζ − z
+�
+¯z
+z dz +
+�
+¯ζ
+ζ · 1{ζ∈Int(�S)}.
+(2.40)
+Recall that f is given by (1.11). Let
+g(w) :=
+�
+f(1/ ¯w)
+f(w) f ′(w).
+(2.41)
+Since f ′(aτ) = 0, the function g(w) has poles only at 0, 1/a, a. We also write
+(2.42)
+hζ(w) :=
+g(w)
+ζ − f(w).
+Using the change of variable z = f(w),
+1
+2πi
+�
+∂ �S
+1
+ζ − z
+�
+¯z
+z dz =
+1
+2πi
+�
+∂D
+1
+ζ − f(w)
+�
+f(1/ ¯w)
+f(w) f ′(w) dw =
+1
+2πi
+�
+∂D
+hζ(w) dw.
+(2.43)
+By the residue calculus, we have
+(2.44)
+Res
+w=0
+�
+hζ(w)
+�
+= 1
+τ ,
+Res
+w=a
+�
+hζ(w)
+�
+= 0.
+Note that ζ = f(w) is equivalent to
+(2.45)
+d(1 − aw)(w − aτ)2 = w(w − a)ζ,
+d = (1 + τ)(1 + 2c)
+2
+,
+which can be rewritten as a cubic equation
+(2.46)
+adw3 − (d + 2a2dτ − ζ)w2 + a(2dτ + a2τ 2d − ζ)w − a2τ 2d = 0.
+For given ζ ∈ C, there exist w(j)
+ζ
+(j = 1, 2, 3) such that f(w(j)
+ζ ) = ζ. Note that by (2.46), we have
+(2.47)
+w(1)
+ζ w(2)
+ζ w(3)
+ζ
+= aτ 2 ∈ (−1, 0).
+Furthermore, since f is a conformal map from Dc onto �Sc, we have the following:
+(1) If ζ ∈ Int(�S), then all w(j)
+ζ ’s are in D;
+(2) If ζ ∈ �Sc, then two of w(j)
+ζ ’s are in D.
+
+EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+13
+By the residue calculus using (1.11) and (2.41), for each j,
+Res
+w=w(j)
+ζ
+�
+hζ(w)
+�
+= −
+g(w(j)
+ζ )
+f ′(w(j)
+ζ )
+= −
+(w(j)
+ζ
+− a)(1 − aτw(j)
+ζ )
+(w(j)
+ζ
+− aτ)(1 − aw(j)
+ζ )
+= d
+ζ
+(aτw(j)
+ζ
+− 1)(w(j)
+ζ
+− aτ)
+w(j)
+ζ
+= d
+ζ
+�
+aτw(j)
+ζ
+− (a2τ 2 + 1) + aτ
+w(j)
+ζ
+�
+,
+(2.48)
+where we have used (2.45). On the other hand, it follows from (2.46) that
+(2.49)
+3
+�
+j=1
+w(j)
+ζ
+= d + 2a2dτ − ζ
+ad
+,
+3
+�
+j=1
+1
+w(j)
+ζ
+= −ζ + 2dτ + a2τ 2d
+aτ 2d
+.
+These relations give rise to
+d
+3
+�
+j=1
+�
+aτw(j)
+ζ
+− (a2τ 2 + 1) + aτ
+w(j)
+ζ
+�
+= dτ + 2a2dτ 2 − τζ − ζ
+τ + 2d + a2τd − 3d(a2τ 2 + 1)
+= −
+�
+τ + 1
+τ
+�
+ζ − c(1 − τ 2).
+(2.50)
+Combining all of the above, we have shown that if ζ ∈ Int(�S),
+(2.51)
+3
+�
+j=1
+Res
+w=w(j)
+ζ
+�
+hζ(w)
+�
+= −
+�
+τ + 1
+τ
+�
+− c(1 − τ 2)
+ζ
+.
+Therefore if ζ ∈ Int(�S), we obtain
+(1 − τ 2)C(ζ) =
+�
+¯ζ
+ζ − 1
+τ − c(1 − τ 2) = (1 − τ 2)∂ζ �Q(ζ).
+(2.52)
+Then by (2.40), the variational equality (1.29) follows.
+Now it remains to show the variational inequality (1.30). Note that by definition, H(ζ) → ∞ as ζ → ∞.
+Suppose that the variational inequality (1.30) does not hold. Then there exists ζ∗ ∈ �Sc such that
+(2.53)
+∂ζH(ζ)|ζ=ζ∗ = ∂ �Q(ζ∗) − C(ζ∗) = 0.
+Recall that if ζ ∈ �Sc, then only one of w(j)
+ζ ’s, say wζ, is in Dc. By combining the above computations, we have
+that for ζ ∈ �Sc,
+(1 − τ 2)
+�
+∂ζ �Q(ζ) − C(ζ)
+�
+=
+�
+¯ζ
+ζ − Res
+w=wζ
+�
+hζ(w)
+�
+=
+�
+¯ζ
+ζ − d
+ζ
+�
+aτwζ − (a2τ 2 + 1) + aτ
+wζ
+�
+.
+(2.54)
+Therefore the identity (2.53) holds if and only if
+(2.55)
+|ζ∗| = d
+�
+aτwζ∗ − (a2τ 2 + 1) + aτ
+wζ∗
+�
+.
+Note that by (1.11),
+−
+d
+f(x)
+�
+aτx − (a2τ 2 + 1) + aτ
+x
+�
+= −
+1
+f(x)
+(1 + τ)(1 + 2c)
+2
+(aτx − 1)(x − aτ)
+x
+= (aτx − 1)(x − a)
+(ax − 1)(x − aτ).
+Therefore if x < 1/(aτ),
+d
+�
+aτx − (a2τ 2 + 1) + aτ
+x
+�
+< τ|f(x)| < |f(x)|.
+From this, we notice that (2.55) does not hold for wζ∗ ∈ R. Furthermore, this implies that the right-hand side
+of (2.55) is real-valued if and only if wζ∗ ∈ ∂D, equivalently, ζ∗ ∈ ∂ �S. This contradicts with the assumption
+that ζ∗ ∈ �Sc. Now the proof is complete.
+□
+
+14
+SUNG-SOO BYUN
+Appendix A. Conformal mapping method: the pre-critical case
+In this appendix, we present the conformal mapping method, which is helpful to derive the candidate of the
+droplet given in terms of the rational function (1.11).
+Proposition A.1. Let τ ∈ (τc, 1). Suppose that �S in (1.18) is simply connected. Let f be a unique conformal
+map (¯Dc, ∞) → (�Sc, ∞), which satisfies
+(A.1)
+f(z) = r1 z + r2 + O
+�1
+z
+�
+,
+z → ∞.
+Then the following holds.
+(i) The conformal map f is a rational function of the form
+(A.2)
+f(z) = r1z + r2 + r3
+z +
+r4
+z − a,
+a ∈ (−1, 0),
+which satisfies
+(A.3)
+f(1/a) = r1
+a + r2 + r3a +
+ar4
+1 − a2 = 0.
+(ii) The parameters rj (j = 1, . . . , 4) are given by
+r1 = 1 + τ
+2
+�
+1 + 2c
+τ
+,
+r2 = 1 + τ
+2τ
+(τ(1 + 2c) + 2τ − 1),
+r3 = 1 + τ
+2
+τ
+�
+τ(1 + 2c),
+r4 = (1 − τ)2(1 + τ)(1 − (1 + 2c)τ)
+2τ
+�
+τ(1 + 2c)
+(A.4)
+and
+(A.5)
+a = −
+1
+�
+τ(1 + 2c)
+.
+Note that the rational function f with the choice of parameters (A.4) corresponds to (1.11). Therefore
+Proposition A.1 gives rise to Theorem 1.4 (ii) under the assumption that �S is simply connected. However, there
+is no general theory characterising the connectivity of the droplet. (Nevertheless, we refer the reader to [49, 48]
+for sharp connectivity bounds of the droplets associated with a class of potentials.) Thus we need to directly
+verify the variational principles as in Subsection 2.2.
+Proof of Proposition A.1 (i). By differentiating the variational equality (1.29), we have
+(A.6)
+∂ζ �Q(ζ) = C(ζ) :=
+� d�µ(z)
+ζ − z ,
+ζ ∈ �S.
+Using (1.14), this can be rewritten as
+(A.7)
+¯ζ = ζ
+�
+(1 − τ 2)
+�
+C(ζ) + c
+ζ
+�
++ τ
+�2
+.
+Therefore the Schwarz function F associated with the droplet �S exists. Furthermore, it is expressed in terms
+of C as
+(A.8)
+F(ζ) = ζ
+�
+(1 − τ 2)
+�
+C(ζ) + c
+ζ
+�
++ τ
+�2
+.
+Note that for z ∈ ∂D,
+(A.9)
+f(1/¯z) = f(z) = f(z)
+�
+(1 − τ 2)
+�
+C(f(z)) +
+c
+f(z)
+�
++ τ
+�2
+.
+Using this, we define f : ¯D\{0} → C by analytic continuation as
+(A.10)
+f(z) := f(1/¯z)
+�
+(1 − τ 2)
+�
+C(f(1/¯z)) +
+c
+f(1/¯z)
+�
++ τ
+�2
+.
+Therefore f has simple poles only at 0, ∞ and the point a ∈ R such that f(1/a) = 0, which leads to (A.2).
+□
+
+EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+15
+Next, we need to specify the constants rj and a. For this, we shall find interrelations among the parameters.
+Lemma A.2. We have
+(A.11)
+r3 = r1τ 2
+and
+(A.12)
+r4 = a(1 − τ 2)
+�
+r2 − 2τ(1 + c)
+�
+.
+Furthermore, we have
+(A.13)
+r2 = r1
+1 + a2τ 2
+1 − a2τ 2
+a2 − 1
+a
++ 2a2(1 − τ 2)τ(1 + c)
+1 − a2τ 2
+.
+Proof. Note that
+(A.14)
+f(1/¯z) = r1
+z + r2 + r3z +
+r4z
+1 − az .
+Therefore, we have
+(A.15)
+1
+f(1/¯z)
+= 1
+r1
+z − r2
+r2
+1
+z2 + r2
+2 − r1r3 − r1r4
+r3
+1
+z3 + O(z4),
+z → 0.
+Since the Cauchy transform C satisfies the asymptotic behaviour
+(A.16)
+C(ζ) = 1
+ζ + O( 1
+ζ2 ),
+ζ → ∞,
+we have
+(A.17)
+C(f(1/¯z)) = 1
+r1
+z + O(z2),
+z → 0.
+Combining these equations with (A.10), we obtain
+f(z) = r1τ 2
+z
++
+�
+r2τ 2 + 2τ(1 − τ 2)(1 + c)
+�
++ O(z),
+z → 0.
+(A.18)
+On the other hand, by using (A.2), we have
+(A.19)
+f(z) = r3
+z +
+�
+r2 − r4
+a
+�
++ O(z),
+z → 0.
+Then by comparing the coefficients in (A.18) and (A.19), we obtain (A.11) and (A.12).
+Note that by (A.3), we have
+(A.20)
+r4 = a2 − 1
+a
+�r1
+a + r2 + r3a
+�
+.
+Then by (A.11), we have
+(A.21)
+r4 = a2 − 1
+a
+�r1
+a + r2 + r1aτ 2�
+= r1
+(1 + a2τ 2)(a2 − 1)
+a2
++ r2
+a2 − 1
+a
+.
+Combining this identity with (A.12), we obtain
+(A.22)
+a(1 − τ 2)r2 − 2a(1 − τ 2)τ(1 + c) = r1
+(1 + a2τ 2)(a2 − 1)
+a2
++ r2
+a2 − 1
+a
+,
+which leads to (A.13).
+□
+Lemma A.3. We have
+(A.23)
+�
+(2 − a2 + a4τ 2)r1 + ar2
+��
+r2 − 2τ(1 + c)
+�
+= (1 − τ 2)c2a(a2 − 1).
+
+16
+SUNG-SOO BYUN
+Proof. Using (A.3), we have
+(A.24)
+1
+f(1/¯z)
+=
+a2(a2 − 1)
+(2 − a2)r1 + ar2 + a4r3
+1
+z − a + O(1),
+z → a.
+Then by (A.10) and (A.11), we obtain
+(A.25)
+r4 = (1 − τ 2)2c2 a2(a2 − 1)
+(2 − a2)r1 + ar2 + a4r3
+=
+(1 − τ 2)2c2 a2(1 − a2)2
+r1(a2τ 2 − 1)(1 − a2)2 + r4a2 .
+Now lemma follows from (A.12).
+□
+Proof of Proposition A.1 (ii). Since �µ is a probability measure, we have
+(A.26)
+1 =
+�
+�S
+1
+2(1 − τ 2)
+1
+|z| dA(z) =
+1
+2πi
+�
+∂ �S
+1
+1 − τ 2
+�
+¯z
+z dz,
+where we have used Green’s formula for the second identity. Using the change of variable z = f(w), where f is
+of the form (A.2), this can be rewritten as
+(A.27)
+1
+2πi
+�
+∂D
+�
+f(1/ ¯w)f(w) f ′(w)
+f(w) dw = 1 − τ 2.
+By Lemma A.2 and (A.2), we have
+f(z) = 1 − az
+z(z − a)
+�
+− r1
+a z2 +
+�a2 − 1
+a2
+r1 − r2
+a
+�
+z − aτ 2r1
+�
+,
+(A.28)
+f(1/¯z) =
+z − a
+z(1 − az)
+�
+− aτ 2r1z2 +
+�a2 − 1
+a2
+r1 − r2
+a
+�
+z − r1
+a
+�
+.
+(A.29)
+Note here that by construction, two zeros of f other than 1/a are contained in the unit disc. Using these
+together with straightforward residue calculus, we obtain
+(A.30)
+Resw=0
+��
+f(1/ ¯w)f(w) f ′(w)
+f(w)
+�
+= (1 + c)(1 − τ 2)
+and
+(A.31)
+Resw=a
+��
+f(1/ ¯w)f(w) f ′(w)
+f(w)
+�
+= −1
+a
+��1 + a2τ 2
+a
+r1 + r2
+��a4τ 2 − a2 + 2
+a
+r1 + r2
+��1/2
+.
+Furthermore, it follows from Lemma A.3 that
+(A.32)
+Resw=a
+��
+f(1/ ¯w)f(w) f ′(w)
+f(w)
+�
+= −c(1 − τ 2).
+Combining (A.27), (A.30) and (A.32), one can notice that the function f has a double zero, which implies that
+(A.33)
+a2 − 1
+a2
+r1 − r2
+a = 2r1τ.
+By solving the system of equations given in Lemmas A.2, A.3 and (A.33), the desired result follows.
+□
+Remark A.4 (The use of higher moments of the equilibrium measure). In a more complicated case, for instance
+for the case with multiple point charges such as (2.7), the mass-one condition (A.26) may not be enough to
+characterise the parameters. In this case, one can further use the higher order asymptotic expansions appearing
+in the above lemmas, which involve the k-th moments of the equilibrium measure; cf. (2.35). Thus in principle,
+one can always find enough (algebraic) interrelations to characterise the parameters appearing in the conformal
+map.
+Remark A.5. For the case τ = 0 and p > 0, it was shown in [12] that if
+c > (1 − p2)2
+4p2
+,
+
+EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+17
+the droplet associated with (2.1) is a simply connected domain whose boundary is given by the image of the
+conformal map
+f(z) = R z −
+κ
+z − q − κ
+q ,
+R = 1 + p2q2
+2pq
+,
+κ = (1 − q2)(1 − p2q2)
+2pq
+.
+Here, q is given by the unique solution of P(q2) = 0, where
+P(x) := x3 −
+�p2 + 4c + 2
+2p2
+�
+x2 +
+1
+2p4
+such that 0 < q < 1 and κ > 0.
+Beyond the case τ = 0, the conformal mapping method described above also works for the potential (2.1) with
+general τ ∈ [0, 1), c ∈ R and p ∈ C under the assumption that the associated droplet is simply connected. Under
+this assumption, one can show that the boundary of the droplet is given by the image of the rational conformal
+map f of the form
+(A.34)
+f(z) = R1 z + R2 + R3
+z +
+R4
+z − q ,
+q ∈ D,
+which satisfies f(1/q) = 0. Furthermore, following the strategy above, one can characterise the coefficients Rj
+(j = 1, . . . , 4) of this rational map as well as the position of the pole q ∈ D.
+However, as previously mentioned, it is far from being obvious to characterise a condition for which the
+droplet is simply connected. Nevertheless, since the radius of curvature of the ellipse (2.3) at the point (1 +
+τ)√1 + c is given by
+(1 − τ)2
+1 + τ
+√
+1 + c,
+one can expect that if
+(A.35)
+p > max
+� 4τ
+1 + τ
+√
+1 + c , (1 + τ)
+√
+1 + c −
+�
+1 − τ 2√c
+�
+then the droplet is a simply connected domain.
+Appendix B. One-dimensional equilibrium measure problem in the Hermitian limit
+In this appendix, we present a proof of (2.10). Let us write
+(B.1)
+V (z) ≡ Vp(z) = z2
+2 − 2c log |z − p|.
+Recall that µV ≡ µVp is the equilibrium measure associated with Vp(x) (x ∈ R).
+We define
+(B.2)
+R(z) :=
+�V ′(z)
+2
+�2
+−
+�
+R
+V ′(z) − V ′(s)
+z − s
+dµV (s).
+By applying Schiffer variations (see e.g. [35, Section 3]), we have
+(B.3)
+R(z) =
+� � dµV (s)
+z − s − V ′(z)
+2
+�2
+,
+z ∈ C \ supp(µV ).
+Combining the asymptotic behaviour
+� dµV (s)
+z − s
+∼ 1
+z ,
+z → ∞,
+with (B.3), we obtain
+(B.4)
+R(z) = 1
+4z2 − (c + 1) − cp
+z + O
+� 1
+z2
+�
+,
+z → ∞.
+On the other hand, since
+V ′(z) = z −
+2c
+z − p,
+V ′(z) − V ′(s)
+z − s
+= 1 +
+2c
+z − p
+1
+s − p,
+
+18
+SUNG-SOO BYUN
+we have
+R(z) = 1
+4
+�
+z −
+2c
+z − p
+�2
+− 1 −
+2c
+z − p
+�
+R
+dµV (s)
+s − p .
+(B.5)
+Thus we obtain
+(B.6)
+R(z) =
+c2
+(z − p)2 + O
+�
+1
+z − p
+�
+,
+z → p.
+In the expression (B.5), one can observe that R is a rational function with a double pole at z = p. Therefore
+it is of the form
+(B.7)
+R(z) = 1
+4z2 + Az2 + Bz + C
+(z − p)2
+for some constants A, B and C. As in the previous subsection, we need to specify these parameters.
+By direct computations, we have
+(B.8)
+R(z) = 1
+4z2 + A + 2Ap + B
+z
++ O
+� 1
+z2
+�
+,
+z → ∞,
+and
+(B.9)
+R(z) = Ap2 + Bp + C
+(z − p)2
++ O
+�
+1
+z − p
+�
+,
+z → p.
+By comparing coefficients in (B.4) and (B.8), we have
+(B.10)
+A = −c − 1,
+−cp = 2Ap + B.
+Similarly, by (B.6) and (B.9),
+(B.11)
+Ap2 + Bp + C = c2.
+By solving these algebraic equations, we obtain
+(B.12)
+B = p(c + 2),
+C = c2 − p2.
+Combining all of the above with (B.7), we have shown that
+R(z) = 1
+4z2 + −(c + 1)z2 + p(c + 2)z + (c2 − p2)
+(z − p)2
+= ((z − p)(z − 2) − 2c)((z − p)(z + 2) − 2c)
+4(z − p)2
+=
+�4
+j=1(z − λj)
+4(z − p)2
+,
+(B.13)
+where λj’s are given by (2.11) and (2.12). Therefore by (B.3), the Stieltjes transform of µV is given by
+� dµV (s)
+z − s
+= V ′(z)
+2
+− R(z)1/2 = z
+2 −
+c
+z − p − 1
+2
+��4
+j=1(z − λj)
+(z − p)2
+.
+(B.14)
+Letting z = x + iε → x ∈ R, we find
+lim
+ε→0+ Im
+�
+dµV (s)
+(x + iε) − s =
+�
+�
+�
+�
+�
+�
+�
+�
+− �4
+j=1(x − λj)
+2|x − p|
+if x ∈ [λ1, λ2] ∪ [λ3, λ4],
+0
+otherwise.
+Now the desired identity (2.10) follows from the Sokhotski-Plemelj inversion formula, see e.g. [39, Section I.4.2].
+Acknowledgements. The author is greatly indebted to Yongwoo Lee for the figures and numerical simulations.
+
+EQUILIBRIUM MEASURE FOR THE QUADRATIC POTENTIALS WITH A POINT CHARGE
+19
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